let D, C be non empty set ; :: thesis: for f, g being PartFunc of C,D st dom f c= dom g & ( for c being Element of C st c in dom f holds
f /. c = g /. c ) holds
f c= g
let f, g be PartFunc of C,D; :: thesis: ( dom f c= dom g & ( for c being Element of C st c in dom f holds
f /. c = g /. c ) implies f c= g )
assume that
A4: dom f c= dom g and
A5: for c being Element of C st c in dom f holds
f /. c = g /. c ; :: thesis: f c= g

now

let x be set ; :: thesis: ( x in dom f implies f . x = g . x )
assume A6: x in dom f ; :: thesis: f . x = g . x
then reconsider x1 = x as Element of C ;
f /. x1 = g /. x1 by A5, A6;
then f . x = g /. x1 by A6, PARTFUN1:def 8;
hence f . x = g . x by A4, A6, PARTFUN1:def 8; :: thesis: verum

end;

hence f c= g by A4, GRFUNC_1:8; :: thesis: verum