
theorem XLMOD02:
  for k,m be Nat st m <> 0 & (k+1) mod m <> 0
  holds (k+1) mod m = (k mod m)+1
proof
  let k,m be Nat;
  assume
C1: m <> 0 & (k+1) mod m <> 0;
  (k mod m)+1 <= m by NAT_D:1,C1,NAT_1:13;
  then
P1: (k mod m)+1-1 <= m-1 by XREAL_1:9;
P2: (k+1) mod m = ((k mod m)+1) mod m by NAT_D:22;
  k mod m < m-1
  proof
    assume not k mod m < m-1;
    then (k+1) mod m = (m-1+1) mod m by XXREAL_0:1,P1,P2
      .= 0 by INT_1:50;
    hence contradiction by C1;
  end;
  then (k mod m)+1 < m-1+1 by XREAL_1:8;
  hence (k+1) mod m = (k mod m)+1 by NAT_D:24,P2;
end;
