let F, G be PartFunc of (REAL m),(Funcs ((REAL m),REAL)); :: thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = diff (f,x) ) implies F = G )
assume that
A7: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) ) and
A8: ( dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = diff (f,x) ) ) ; :: thesis: F = G
now :: thesis: for x being Element of REAL m st x in dom F holds
F /. x = G /. x
let x be Element of REAL m; :: thesis: ( x in dom F implies F /. x = G /. x )
assume A9: x in dom F ; :: thesis: F /. x = G /. x
then F /. x = diff (f,x) by A7;
hence F /. x = G /. x by A7, A8, A9; :: thesis: verum
end;
hence F = G by A7, A8, PARTFUN2:1; :: thesis: verum