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;
reserve L for non empty ZeroStr;
reserve x for Element of L;
reserve p,q for Polynomial of F_Real;

theorem
  for r,s being Element of F_Real holds poly_diff(<%r,s%>) = <%s%>
  proof
    let r,s be Element of F_Real;
    let n be Element of NAT;
A1: (poly_diff(<%r,s%>)).n = <%r,s%>.(n+1) * (n+1) by Def5;
    per cases;
    suppose
A2:   n = 0;
      hence (poly_diff(<%r,s%>)).n = s by A1,POLYNOM5:38
      .= <%s%>.n by A2,POLYNOM5:32;
    end;
    suppose n <> 0;
      then
A3:   0+1 <= n by NAT_1:13;
      1+1 <= n+1 by A3,XREAL_1:6;
      then <%r,s%>.(n+1) = 0.F by POLYNOM5:38;
      hence thesis by A1,A3,POLYNOM5:32;
    end;
  end;
