let X, Y be set ; :: thesis: for f, g being PartFunc of X,Y st ( Y = {} implies X = {} ) & f tolerates g holds
ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h )
let f, g be PartFunc of X,Y; :: thesis: ( ( Y = {} implies X = {} ) & f tolerates g implies ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h ) )
assume that
A1: ( Y = {} implies X = {} ) and
A2: f tolerates g ; :: thesis: ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h )

now

per cases ( Y = {} or Y <> {} ) ;

suppose A3: Y = {} ; :: thesis: ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h )

then ( f tolerates <:{},X,Y:> & g tolerates <:{},X,Y:> ) ;
hence ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h ) by A1, A3; :: thesis: verum

end;

suppose A4: Y <> {} ; :: thesis: ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h )

consider y being Element of Y;
defpred S₁[ set , set ] means ( ( $1 in dom f implies $2 = f . $1 ) & ( $1 in dom g implies $2 = g . $1 ) & ( not $1 in dom f & not $1 in dom g implies $2 = y ) );
A5: for x being set st x in X holds
ex y9 being set st S₁[x,y9]

proof

let x be set ; :: thesis: ( x in X implies ex y9 being set st S₁[x,y9] )
assume x in X ; :: thesis: ex y9 being set st S₁[x,y9]

now

per cases ( ( x in dom f & x in dom g ) or ( x in dom f & not x in dom g ) or ( not x in dom f & x in dom g ) or ( not x in dom f & not x in dom g ) ) ;

suppose A6: ( x in dom f & x in dom g ) ; :: thesis: ex y9 being set st
( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )

take y9 = f . x; :: thesis: ( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )
thus ( x in dom f implies y9 = f . x ) ; :: thesis: ( ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )
x in (dom f) /\ (dom g) by A6, XBOOLE_0:def 4;
hence ( x in dom g implies y9 = g . x ) by A2, Def6; :: thesis: ( not x in dom f & not x in dom g implies y9 = y )
thus ( not x in dom f & not x in dom g implies y9 = y ) by A6; :: thesis: verum

end;

suppose A7: ( x in dom f & not x in dom g ) ; :: thesis: ex y9 being set st
( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )

take y9 = f . x; :: thesis: ( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )
thus ( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) ) by A7; :: thesis: verum

end;

suppose A8: ( not x in dom f & x in dom g ) ; :: thesis: ex y9 being set st
( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )

take y9 = g . x; :: thesis: ( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) )
thus ( ( x in dom f implies y9 = f . x ) & ( x in dom g implies y9 = g . x ) & ( not x in dom f & not x in dom g implies y9 = y ) ) by A8; :: thesis: verum

end;

suppose ( not x in dom f & not x in dom g ) ; :: thesis: ex y9 being set st S₁[x,y9]

hence ex y9 being set st S₁[x,y9] ; :: thesis: verum

end;

end;

end;

hence ex y9 being set st S₁[x,y9] ; :: thesis: verum

end;

A9: for x, y1, y2 being set st x in X & S₁[x,y1] & S₁[x,y2] holds
y1 = y2 ;
consider h being Function such that
A10: dom h = X and
A11: for x being set st x in X holds
S₁[x,h . x] from FUNCT_1:sch 2(A9, A5);
rng h c= Y

proof

let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in rng h or z in Y )
assume z in rng h ; :: thesis: z in Y
then consider x being set such that
A12: x in dom h and
A13: z = h . x by FUNCT_1:def 5;

per cases ( ( x in dom f & x in dom g ) or ( x in dom f & not x in dom g ) or ( not x in dom f & x in dom g ) or ( not x in dom f & not x in dom g ) ) ;

suppose A14: ( x in dom f & x in dom g ) ; :: thesis: z in Y

then z = f . x by A11, A13;
hence z in Y by A14, Th27; :: thesis: verum

end;

suppose A15: ( x in dom f & not x in dom g ) ; :: thesis: z in Y

then z = f . x by A11, A13;
hence z in Y by A15, Th27; :: thesis: verum

end;

suppose A16: ( not x in dom f & x in dom g ) ; :: thesis: z in Y

then z = g . x by A11, A13;
hence z in Y by A16, Th27; :: thesis: verum

end;

suppose ( not x in dom f & not x in dom g ) ; :: thesis: z in Y

then z = y by A10, A11, A12, A13;
hence z in Y by A4; :: thesis: verum

end;

end;

end;

then reconsider h9 = h as PartFunc of X,Y by A10, RELSET_1:11;
A17: f tolerates h

proof

let x be set ; :: according to PARTFUN1:def 6 :: thesis: ( x in (dom f) /\ (dom h) implies f . x = h . x )
assume x in (dom f) /\ (dom h) ; :: thesis: f . x = h . x
then x in dom f by XBOOLE_0:def 4;
hence f . x = h . x by A11; :: thesis: verum

end;

A18: g tolerates h

proof

let x be set ; :: according to PARTFUN1:def 6 :: thesis: ( x in (dom g) /\ (dom h) implies g . x = h . x )
assume x in (dom g) /\ (dom h) ; :: thesis: g . x = h . x
then x in dom g by XBOOLE_0:def 4;
hence g . x = h . x by A11; :: thesis: verum

end;

h9 is total by A10, Def4;
hence ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h ) by A17, A18; :: thesis: verum

end;

end;

end;

hence ex h being PartFunc of X,Y st
( h is total & f tolerates h & g tolerates h ) ; :: thesis: verum