reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th80:
  a is odd implies multiples(2) /\ {a,a+1} = {a+1}
  proof
    set M = multiples(2);
    assume
A1: a is odd;
    thus M /\ {a,a+1} c= {a+1}
    proof
      let x be object;
      assume
A2:   x in M /\ {a,a+1};
      then per cases by TARSKI:def 2;
      suppose x = a+1;
        hence thesis by TARSKI:def 1;
      end;
      suppose
A3:     x = a;
        reconsider x as Multiple of 2 by A2,Th61;
        2 divides x by Def15;
        hence thesis by A3,A1;
      end;
    end;
    let x be object;
    assume x in {a+1};
    then
A4: x = a+1 by TARSKI:def 1;
A5: a+1 in M by A1,Th76;
    a+1 in {a,a+1} by TARSKI:def 2;
    hence thesis by A4,A5,XBOOLE_0:def 4;
  end;
