reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem Th40:
  p1 <> p2 implies 2 <= p1 & 3 <= p2 or 3 <= p1 & 2 <= p2
  proof
    assume
A1: p1 <> p2;
    assume 2 > p1 or 3 > p2;
    then per cases;
    suppose p1 < 1+1;
      then p1 <= 1 by NAT_1:13;
      hence thesis by INT_2:def 4;
    end;
    suppose
A2:   p2 < 3;
      1 < p2 by INT_2:def 4;
      then
A3:   1+1 <= p2 by NAT_1:13;
      1 < p1 by INT_2:def 4;
      then
A4:   1+1 <= p1 by NAT_1:13;
      2 = 3-1;
      then p2 = 0 or ... or p2 = 2 by A2,NUMBER02:7;
      then 2 < p1 by A1,A4,INT_2:def 4,XXREAL_0:1;
      then 2+1 <= p1 by NAT_1:13;
      hence thesis by A3;
    end;
  end;
