 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th3:
  for f be Element of the carrier of Polynom-Ring R holds
    rng f = f.:(Support f) \/ {0.R}
     proof
       let f be Element of the carrier of Polynom-Ring R;
A1:    dom f = NAT by FUNCT_2:def 1;
A2:    rng f = f.:(dom f) by RELAT_1:113
       .= f.:(Support f \/ (NAT \ (Support f))) by A1,XBOOLE_1:45
       .= f.:(Support f) \/ f.:(NAT \ (Support f)) by RELAT_1:120;
A3:    for y be object st y in f.:(NAT \ (Support f)) holds y in {0.R}
       proof
         let y be object;
         assume y in f.:(NAT \ (Support f)); then
         consider x being object such that
A5:      x in dom f & x in NAT \ (Support f) & y = f.x by FUNCT_1:def 6;
         f.x = 0.R
         proof
           assume f.x <> 0.R; then
           x in Support f by A5,POLYNOM1:def 4;
           hence contradiction by A5, XBOOLE_0:def 5;
         end;
         hence thesis by TARSKI:def 1,A5;
       end;
       for y be object st y in {0.R} holds y in f.:(NAT \ (Support f))
       proof
         let y be object;
         assume
A9:      y in {0.R};
         0.R in f.:(NAT \ (Support f))
         proof
A10:       f.(len f) = 0.R by ALGSEQ_1:8; then
           not len f in (Support f) by POLYNOM1:def 4; then
           len f in NAT \ (Support f) by XBOOLE_0:def 5;
           hence thesis by A10,A1,FUNCT_1:def 6;
         end;
         hence thesis by A9,TARSKI:def 1;
       end;
       hence thesis by A2,A3,TARSKI_0:2;
     end;
