reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;

theorem
  (f-r)`| = f`|
  proof
    reconsider s = r as Element of REAL by XREAL_0:def 1;
    set g = REAL --> s;
A1: g`| = REAL --> 0 by Th11;
A2: dom(f`|) = REAL by FUNCT_2:def 1;
    dom f = REAL by FUNCT_2:def 1;
    then f-r = f-g by Th4;
    hence (f-r)`| = f`|-g`| by Th15
    .= f`| by A1,A2,Th7;
  end;
