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|
[
[ BeginEnv "proposition"
, Label "trivial"
, BeginEnv "math"
, Variable "x"
, Symbol "="
, Variable "x"
, EndEnv "math"
, Symbol "."
, EndEnv "proposition"
]
,
[ BeginEnv "proposition"
, Label "irrelevant"
, BeginEnv "math"
, Variable "z"
, Symbol "="
, Variable "z"
, EndEnv "math"
, Symbol "."
, EndEnv "proposition"
]
,
[ BeginEnv "proposition"
, Label "alsotrivial"
, BeginEnv "math"
, Variable "y"
, Symbol "="
, Variable "y"
, EndEnv "math"
, Symbol "."
, EndEnv "proposition"
]
,
[ BeginEnv "proof"
, BeginEnv "align*"
, Variable "y"
, Symbol "&="
, Variable "y"
, Symbol "&="
, Variable "y"
, BeginEnv "text"
, Word "by"
, Ref
( "trivial" :| [] )
, EndEnv "text"
, EndEnv "align*"
, EndEnv "proof"
]
,
[ BeginEnv "proposition"
, Label "trivial_biconditionals"
, BeginEnv "math"
, Variable "y"
, Symbol "="
, Variable "y"
, EndEnv "math"
, Symbol "."
, EndEnv "proposition"
]
,
[ BeginEnv "proof"
, BeginEnv "align*"
, Variable "y"
, Symbol "="
, Variable "y"
, Symbol "&"
, Command "iff"
, Command "top"
, Symbol "&"
, Command "iff"
, Variable "y"
, Symbol "="
, Variable "y"
, BeginEnv "text"
, Word "by"
, Ref
( "trivial" :| [] )
, EndEnv "text"
, EndEnv "align*"
, EndEnv "proof"
]
,
[ BeginEnv "proposition"
, Label "bounded_calc"
, Word "for"
, Word "all"
, BeginEnv "math"
, Variable "x"
, Command "in"
, Variable "A"
, EndEnv "math"
, Word "we"
, Word "have"
, BeginEnv "math"
, Variable "x"
, Symbol "="
, Variable "x"
, EndEnv "math"
, Symbol "."
, EndEnv "proposition"
]
,
[ BeginEnv "proof"
, Word "for"
, Word "all"
, BeginEnv "math"
, Variable "x"
, Command "in"
, Variable "A"
, EndEnv "math"
, Word "we"
, Word "have"
, BeginEnv "align*"
, Variable "x"
, Symbol "&="
, Variable "x"
, EndEnv "align*"
, EndEnv "proof"
]
,
[ BeginEnv "proposition"
, Label "bounded_calc_such_that"
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, Variable "x"
, Command "in"
, Variable "A"
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, BeginEnv "math"
, Variable "x"
, Symbol "="
, Variable "x"
, EndEnv "math"
, Word "we"
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, BeginEnv "math"
, Variable "x"
, Symbol "="
, Variable "x"
, EndEnv "math"
, Symbol "."
, EndEnv "proposition"
]
,
[ BeginEnv "proof"
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, BeginEnv "math"
, Variable "x"
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, Word "such"
, Word "that"
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, Word "we"
, Word "have"
, BeginEnv "align*"
, Variable "x"
, Symbol "&="
, Variable "x"
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, EndEnv "proof"
]
]
|
