let D, C be non empty set ; :: thesis: for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max- F, max- G are_fiberwise_equipotent
let F be PartFunc of D,REAL ; :: thesis: for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max- F, max- G are_fiberwise_equipotent
let G be PartFunc of C,REAL ; :: thesis: ( F,G are_fiberwise_equipotent implies max- F, max- G are_fiberwise_equipotent )
assume A1: F,G are_fiberwise_equipotent ; :: thesis: max- F, max- G are_fiberwise_equipotent
A2: ( rng F c= REAL & rng G c= REAL & rng (max- F) c= REAL & rng (max- G) c= REAL ) ;
set li = right_closed_halfline 0 ;

now

let r be Real; :: thesis: card (Coim (max- F),r) = card (Coim (max- G),r)

now

per cases ( 0 < r or r = 0 or r < 0 ) ;

case 0 < r ; :: thesis: card (Coim (max- F),r) = card (Coim (max- G),r)

then ( Coim F,(- r) = (max- F) " {r} & Coim G,(- r) = (max- G) " {r} ) by Th41;
hence card (Coim (max- F),r) = card (Coim (max- G),r) by A1, CLASSES1:def 9; :: thesis: verum

end;

case A3: r = 0 ; :: thesis: card ((max- F) " {r}) = card ((max- G) " {r})

( F " (right_closed_halfline 0 ) = (max- F) " {0 } & G " (right_closed_halfline 0 ) = (max- G) " {0 } ) by Th42;
hence card ((max- F) " {r}) = card ((max- G) " {r}) by A1, A3, CLASSES1:86; :: thesis: verum

end;

case A4: r < 0 ; :: thesis: card ((max- F) " {r}) = card ((max- G) " {r})

now

assume r in rng (max- F) ; :: thesis: contradiction
then consider d being Element of D such that
A5: ( d in dom (max- F) & (max- F) . d = r ) by PARTFUN1:26;
thus contradiction by A4, A5, Th43; :: thesis: verum

end;

then A6: (max- F) " {r} = {} by Lm2;

now

assume r in rng (max- G) ; :: thesis: contradiction
then consider c being Element of C such that
A7: ( c in dom (max- G) & (max- G) . c = r ) by PARTFUN1:26;
thus contradiction by A4, A7, Th43; :: thesis: verum

end;

hence card ((max- F) " {r}) = card ((max- G) " {r}) by A6, Lm2; :: thesis: verum

end;

end;

end;

hence card (Coim (max- F),r) = card (Coim (max- G),r) ; :: thesis: verum

end;

hence max- F, max- G are_fiberwise_equipotent by A2, CLASSES1:87; :: thesis: verum