
theorem pr20:
for S being Ring,
    R being Subring of S
for x1,x2 being Element of S, y1,y2 being Element of R
st x1 = y1 & x2 = y2 holds <%x1,x2%> = <%y1,y2%>
proof
let S be Ring, R be Subring of S;
let x1,x2 be Element of S, y1,y2 be Element of R;
assume AS: x1 = y1 & x2 = y2;
set p = <%x1,x2%>, q = <%y1,y2%>;
now let n be Element of NAT;
  per cases;
  suppose A: n is trivial;
    per cases by A,NAT_2:def 1;
    suppose B: n = 0;
      hence p.n = y1 by AS,POLYNOM5:38 .= q.n by B,POLYNOM5:38;
      end;
    suppose B: n = 1;
      hence p.n = y2 by AS,POLYNOM5:38 .= q.n by B,POLYNOM5:38;
      end;
    end;
  suppose A: n is non trivial;
    hence p.n = 0.S by NAT_2:29,POLYNOM5:38
             .= 0.R by C0SP1:def 3
             .= q.n by A,NAT_2:29,POLYNOM5:38;
    end;
  end;
hence thesis;
end;
