reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

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;
