reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;

theorem Th35:
  n+m = n +^ m
proof
  defpred P[Nat] means n+$1 = n +^ $1;
A1: for m st P[m] holds P[m+1]
  proof
    let m such that
A2: P[m];
    thus n+(m+1) = Segm(n+m+1)
      .= succ Segm(n + m) by NAT_1:38
      .= succ(n +^ m) by A2
      .= n +^ succ Segm m by ORDINAL2:28
      .= n +^ Segm (m+1) by NAT_1:38
      .= n +^ (m+1);
  end;
A3: P[0] by ORDINAL2:27;
  for m holds P[m] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
