let S, T be non empty RelStr ; :: thesis: for f being Function of S,T st f is isomorphic holds
for g being Function of T,S st g = f " holds
g is isomorphic
let f be Function of S,T; :: thesis: ( f is isomorphic implies for g being Function of T,S st g = f " holds
g is isomorphic )
assume A1: f is isomorphic ; :: thesis: for g being Function of T,S st g = f " holds
g is isomorphic
then A2: ex h being Function of T,S st
( h = f " & h is monotone ) by Def38;
let g be Function of T,S; :: thesis: ( g = f " implies g is isomorphic )
assume A3: g = f " ; :: thesis: g is isomorphic
per cases ( ( not T is empty & not S is empty ) or T is empty or S is empty ) ;
:: according to WAYBEL_0:def 38
case ( not T is empty & not S is empty ) ; :: thesis: ( g is one-to-one & g is monotone & ex g being Function of S,T st
( g = g " & g is monotone ) )
thus ( g is one-to-one & g is monotone ) by A1, A2, A3, FUNCT_1:40; :: thesis: ex g being Function of S,T st
( g = g " & g is monotone )
(f ") " = f by A1, FUNCT_1:43;
hence ex g being Function of S,T st
( g = g " & g is monotone ) by A1, A3; :: thesis: verum
end;
case ( T is empty or S is empty ) ; :: thesis: ( T is empty & S is empty )
hence ( T is empty & S is empty ) ; :: thesis: verum
end;
end;