let C, D be non empty set ; :: thesis: for f, g being PartFunc of C,D st dom f = dom g & ( for c being Element of C st c in dom f holds
f /. c = g /. c ) holds
f = g
let f, g be PartFunc of C,D; :: thesis: ( dom f = dom g & ( for c being Element of C st c in dom f holds
f /. c = g /. c ) implies f = g )
assume that
A1: dom f = dom g and
A2: for c being Element of C st c in dom f holds
f /. c = g /. c ; :: thesis: f = g

now :: thesis: for x being object st x in dom f holds
f . x = g . x

let x be object ; :: thesis: ( x in dom f implies f . x = g . x )
assume A3: x in dom f ; :: thesis: f . x = g . x
then reconsider y = x as Element of C ;
f /. y = g /. y by A2, A3;
then f . y = g /. y by A3, PARTFUN1:def 6;
hence f . x = g . x by A1, A3, PARTFUN1:def 6; :: thesis: verum

end;

hence f = g by A1; :: thesis: verum