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(#)(f2^))|X by Th31
    .= (f1|X) (#) (f2^|X) by Th45
    .= (f1|X) (#) ((f2|X)^) by Th46
    .= (f1|X)/(f2|X) by Th31;
  thus (f1/f2)|X = (f1(#)(f2^))|X by Th31
    .= (f1|X) (#) (f2^) by Th45
    .= (f1|X)/f2 by Th31;
  thus (f1/f2)|X = (f1(#)(f2^))|X by Th31
    .= f1 (#) (f2^)|X by Th45
    .= f1 (#) ((f2|X)^) by Th46
    .= f1/(f2|X) by Th31;
end;
