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

theorem Th12:
  for x,o being Element of REAL st o = 0 holds *(x,o) = 0
proof
  let x,o being Element of REAL such that
A1: o = 0;
  per cases;
  suppose
    x in REAL+;
    then reconsider x9 = x, y9 = 0 as Element of REAL+ by ARYTM_2:20;
    0 = x9 *' y9 by ARYTM_2:4;
    hence thesis by A1,Def2;
  end;
  suppose
A2: not x in REAL+;
    not o in [:{{}},REAL+:] by A1,Th5,ARYTM_2:20,XBOOLE_0:3;
    hence thesis by A1,A2,Def2;
  end;
end;
