reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

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|X)+ (-f2)|X by Th44
    .= (f1|X) - (f2|X) by Th46;
  thus (f1-f2)|X = (f1|X) - f2 by Th44;
  thus (f1-f2)|X = f1+ (-f2)|X by Th44
    .= f1 - (f2|X) by Th46;
end;
