reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 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 Th51
    .= (f1|X) - (f2|X) by Th53;
  thus (f1-f2)|X = (f1|X)+ -f2 by Th51
    .= (f1|X) - f2;
  thus (f1-f2)|X = f1+ (-f2)|X by Th51
    .= f1 - (f2|X) by Th53;
end;
