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 Th59:
  for p,q being Prime, a,b being Nat st a <> 1 & b <> 1 & p*q = a*b holds
  p = a & q = b or p = b & q = a
  proof
    let p,q be Prime, a,b be Nat such that
A1: a <> 1 and
A2: b <> 1 and
A3: p*q = a*b and
A4: p <> a or q <> b;
    a divides a*b;
    then per cases by A1,A3,GR_CY_3:1;
    suppose
A5:   a = p;
      then q <> b by A4;
      hence thesis by A3,A5,XCMPLX_1:5;
    end;
    suppose a = q;
      hence thesis by A3,XCMPLX_1:5;
    end;
    suppose
A6:   a = p*q;
      then a*1 = a*b by A3;
      hence thesis by A2,A6,XCMPLX_1:5;
    end;
  end;
