
theorem YZ:
for R being non degenerated Ring
for n being Ordinal
for p being Polynomial of n,R holds Support(LM p) c= Support p
proof
let R be non degenerated Ring, n be Ordinal, p be Polynomial of n,R;
now let o be object;
  assume A: o in Support(LM p);
  then reconsider b = o as Element of Bags n;
  B: (LM p).b <> 0.R by A,POLYNOM1:def 4;
  D: Lt p is Element of Bags n by PRE_POLY:def 12;
  C: dom 0_(n,R) = Bags n by FUNCT_2:def 1;
  E: LM p = 0_(n,R) +* (Lt p,LC p) by POLYNOM7:def 4
         .= 0_(n,R) +* (Lt p .--> LC p) by C,D,FUNCT_7:def 3;
  per cases;
  suppose H: b in dom(Lt p .--> LC p); then
    I: b = Lt p by TARSKI:def 1;
    (LM p).b = (Lt p .--> LC p).b by H,E,FUNCT_4:13
            .= LC p by I,FUNCOP_1:72;
    hence o in Support p by B,I,POLYNOM1:def 4;
    end;
  suppose not b in dom(Lt p .--> LC p); then
    (LM p).b = (0_(n,R)).b by E,FUNCT_4:11 .= 0.R by POLYNOM1:22;
    hence o in Support p by A,POLYNOM1:def 4;
    end;
  end;
hence thesis;
end;
