theorem Th14:
  (0.REAL n)+*(x,0) = 0.REAL n
  proof
    set p = (0.REAL n)+*(x,0);
A1: dom p = Seg n by FINSEQ_1:89;
A2: dom ((0.REAL n)+*(x,0)) = dom (0.REAL n) by FUNCT_7:30;
A3: dom (0.REAL n) = Seg n;
    now
      let z be object;
      assume
A4:   z in dom p;
      per cases;
      suppose z = x;
        hence p.z = 0 by A1,A3,A4,FUNCT_7:31
        .= (0.REAL n).z;
      end;
      suppose z <> x;
        hence p.z = (0.REAL n).z by FUNCT_7:32;
      end;
    end;
    hence thesis by A2;
  end;
