 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 lem8:
  for i1,i2,i3,i4, c be Nat st i1 <= c & i2 <= c & i3 <= c & i4 <= c holds
    i1+i2 + i3 + i4 < 4*c or (i1 = c & i2 = c & i3 = c & i4 = c)
    proof
      let i1,i2,i3,i4, c be Nat;
      assume that
A1:   i1 <= c and
A2:   i2 <= c and
A3:   i3 <= c and
A4:   i4 <= c;
      per cases by A1,A2, A3,A4,lem7;
      suppose i1+i2 < 2*c & i3+i4 < 2*c; then
        (i1 + i2) + (i3 + i4) < 2*c +2*c by XREAL_1:8;
        hence thesis;
      end;
      suppose i1+i2 < 2*c & i3 = c & i4 = c; then
        (i1+i2) + (i3 +i4) < 2*c + 2*c by XREAL_1:8;
        hence thesis;
      end;
      suppose (i1 = c & i2 = c) & i3+i4 < 2*c; then
        (i1+i2) + (i3 +i4) < 2*c + 2*c by XREAL_1:8;
        hence thesis;
      end;
      suppose i1 = c & i2 = c & i3 = c & i4 = c;
        hence thesis;
      end;
    end;
