 reserve o for object;
 reserve F for non almost_trivial Field;
 reserve x,a for Element of F;
reserve n for non zero Nat;

theorem Th21:
   for R being Ring, p being Polynomial of R for r being Real holds p <> r
   proof
     let R be Ring, p be Polynomial of R; let r be Real;
A1:  r in REAL+ \/ [:{0},REAL+:] \ {[0,0]} by XREAL_0:def 1,NUMBERS:def 1;
     now assume
A2:    p = r; then
       not r in REAL+ by Lem4; then
       r in [:{0},REAL+:] by A1,XBOOLE_0:def 3; then
       consider x,y being object such that
A3:    x in {0} & y in REAL+ & r = [x,y] by ZFMISC_1:def 2;
       dom p = NAT by FUNCT_2:def 1; then
       [1,p.1] in p by FUNCT_1:def 2; then
A4:    [1,p.1] in {{x,y},{x}} by A3,A2,TARSKI:def 5;
       per cases by A4,TARSKI:def 2;
         suppose [1,p.1] = {x,y}; then
A5:        {{1,p.1},{1}} = {x,y} by TARSKI:def 5;
           x in {x,y} by TARSKI:def 2; then
           per cases by A5,TARSKI:def 2;
             suppose x = {1,p.1}; then
               x <> {};
               hence contradiction by A3,TARSKI:def 1;
             end;
             suppose x = {1}; then
               x <> {};
               hence contradiction by A3,TARSKI:def 1;
             end;
           end;
           suppose [1,p.1] = {x};
             hence contradiction by A3,TARSKI:def 1,CARD_1:49;
           end;
         end;
         hence thesis;
       end;
