 reserve i,j, k,v, w for Nat;
 reserve j1,j2, m, n, s, t, x, y for Integer;
 reserve p for odd Prime;
 reserve a for Real;
 reserve b for Integer;

theorem lem7:
  for i1,i2, c be Nat st i1 <= c & i2 <= c holds
    i1+i2 < 2*c or (i1 = c & i2 = c)
  proof
    let i1,i2, c be Nat;
    assume that
A1: i1 <= c and
A2: i2 <= c;
    i1 in [.0,c.] by A1,XXREAL_1:1; then
A3: i1 in [.0,c.[ or i1 = c by XXREAL_1:7;
    i2 in [.0,c.] by A2,XXREAL_1:1; then
A4: i2 in [.0,c.[ or i2 = c by XXREAL_1:7;
    per cases by A3,XXREAL_1:3,A4;
    suppose i1 = c & i2 = c;
      hence thesis;
    end;
    suppose 0 <= i1 & i1 < c & 0 <= i2 & i2 < c; then
      i1 + i2 < c + c by XREAL_1:8;
      hence thesis;
    end;
    suppose 0 <= i1 & i1 < c & i2 = c; then
      i1 + i2 < c + c by XREAL_1:8;
      hence thesis;
    end;
    suppose 0 <= i2 & i2 < c & i1 = c; then
      i1 + i2 < c + c by XREAL_1:8;
      hence thesis;
    end;
  end;
