let X, Y be set ; :: thesis: for f being PartFunc of X,Y st ( Y = {} implies X = {} ) holds
ex g being Function of X,Y st
for x being set st x in dom f holds
g . x = f . x
let f be PartFunc of X,Y; :: thesis: ( ( Y = {} implies X = {} ) implies ex g being Function of X,Y st
for x being set st x in dom f holds
g . x = f . x )
assume A1: ( Y = {} implies X = {} ) ; :: thesis: ex g being Function of X,Y st
for x being set st x in dom f holds
g . x = f . x

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

suppose Y = {} ; :: thesis: ex g being Function of X,Y st
for x being set st x in dom f holds
g . x = f . x

then reconsider g = f as Function of X,Y by A1;
take g ; :: thesis: for x being set st x in dom f holds
g . x = f . x
thus for x being set st x in dom f holds
g . x = f . x ; :: thesis: verum

end;

suppose A4: Y <> {} ; :: thesis: ex g being Function of X,Y st
for x being set st x in dom f holds
g . x = f . x

consider y being Element of Y;
defpred S₁[ set ] means $1 in dom f;
deffunc H₁( set ) -> set = f . $1;
deffunc H₂( set ) -> Element of Y = y;
A5: for x being set st x in X holds
( ( S₁[x] implies H₁(x) in Y ) & ( not S₁[x] implies H₂(x) in Y ) ) by A4, PARTFUN1:27;
consider g being Function of X,Y such that
A6: for x being set st x in X holds
( ( S₁[x] implies g . x = H₁(x) ) & ( not S₁[x] implies g . x = H₂(x) ) ) from FUNCT_2:sch 5(A5);
take g ; :: thesis: for x being set st x in dom f holds
g . x = f . x
thus for x being set st x in dom f holds
g . x = f . x by A6; :: thesis: verum

end;

end;