reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem Th19:
  for x,y st y = 1 holds *(x,y) = x
proof
  let x,y such that
A1: y = 1;
  per cases;
  suppose
    x = 0;
    hence thesis by Th12;
  end;
  suppose
A2: x <> 0;
A3: now
      assume
A4:   inv x = 0;
      thus 1 = *(x, inv x) by A2,Def4
        .= 0 by A4,Th12;
    end;
A5: ex x9,y9 being Element of REAL+ st y = x9 & y = y9 & *(y,y) = x9 *' y9
    by A1,Def2,ARYTM_2:20;
A6: *(x,inv x) = 1 by A2,Def4;
    *(*(x,y), inv x) = *(*(x,inv x), y) by Th13
      .= *(y,y) by A1,A2,Def4
      .= 1 by A1,A5,ARYTM_2:15;
    hence thesis by A3,A6,Th18;
  end;
end;
