let IT1, IT2 be Function; :: thesis: ( dom IT1 = (dom f) \/ (dom g) & ( for x being object st x in (dom f) \/ (dom g) holds
IT1 . x = (f . x) \/ (g . x) ) & dom IT2 = (dom f) \/ (dom g) & ( for x being object st x in (dom f) \/ (dom g) holds
IT2 . x = (f . x) \/ (g . x) ) implies IT1 = IT2 )
assume that
A4: dom IT1 = (dom f) \/ (dom g) and
A5: for x being object st x in (dom f) \/ (dom g) holds
IT1 . x = (f . x) \/ (g . x) and
A6: dom IT2 = (dom f) \/ (dom g) and
A7: for x being object st x in (dom f) \/ (dom g) holds
IT2 . x = (f . x) \/ (g . x) ; :: thesis: IT1 = IT2

now :: thesis: for x being object st x in dom IT1 holds
IT1 . x = IT2 . x

let x be object ; :: thesis: ( x in dom IT1 implies IT1 . x = IT2 . x )
assume A8: x in dom IT1 ; :: thesis: IT1 . x = IT2 . x
IT1 . x = (f . x) \/ (g . x) by A4, A5, A8;
hence IT1 . x = IT2 . x by A4, A7, A8; :: thesis: verum

end;

hence IT1 = IT2 by A4, A6, FUNCT_1:2; :: thesis: verum