reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem
  (f1-f2)|X = f1|X - f2|X & (f1-f2)|X = f1|X - f2 &(f1-f2)|X = f1 - f2|X
proof
  thus (f1-f2)|X = (f1+-f2)|X by Th25
    .= (f1|X)+ (-f2)|X by Th27
    .= (f1|X)+ -(f2|X) by Th29
    .= (f1|X) - (f2|X) by Th25;
  thus (f1-f2)|X = (f1+-f2)|X by Th25
    .= (f1|X)+ -f2 by Th27
    .= (f1|X) - f2 by Th25;
  thus (f1-f2)|X = (f1+-f2)|X by Th25
    .= f1+ (-f2)|X by Th27
    .= f1 +- (f2|X) by Th29
    .= f1 - (f2|X) by Th25;
end;
