reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem
  for O be Ordinal, R be right_zeroed add-associative right_complementable
      right_unital distributive non trivial doubleLoopStr
    for p be Polynomial of O,R, b be bag of O
  holds Support(p - Monom(p.b,b)) = Support(p)\{b}
proof
  let O be Ordinal, R be right_zeroed add-associative right_complementable
  right_unital distributive non trivial doubleLoopStr;
  let p be Polynomial of O,R, b be bag of O;
  set M=Monom(p.b,b);
  per cases;
  suppose
A1: p.b =0.R;
    then M = 0_(O,R) by Th25;
    then
A2: p - M = p by POLYRED:4;
    not b in Support(p) by A1,POLYNOM1:def 3;
    hence thesis by A2,ZFMISC_1:57;
   end;
   suppose p.b<>0.R;
A3:  dom 0_(O,R) = Bags O by FUNCT_2:def 1;
A4:  dom p = Bags O =dom (p - M) by FUNCT_2:def 1;
     thus Support(p - M) c= Support(p)\{b}
     proof
       let x;
       assume
A5:    x in Support(p-M);
       then reconsider x as bag of O;
A6:    (p - M).x = (p+ (-M)).x by POLYNOM1:def 7
       .= (p.x) +((-M).x) by POLYNOM1:15
       .= (p.x) +(-(M.x)) by POLYNOM1:17;
A7:    x<>b
       proof
         assume x= b;
         then M.x = p.x by A5,A3,FUNCT_7:31;
         then (p.x) -(M.x) = 0.R by RLVECT_1:15;
         hence thesis by A6,A5,POLYNOM1:def 3;
       end;
       M.x = 0_(O,R).x by A7,FUNCT_7:32;
       then M.x = 0.R by POLYNOM1:22;
       then p.x <>0.R by A6,A5,POLYNOM1:def 3;
       then x in Support(p) by A4,A5,POLYNOM1:def 3;
       hence thesis by A7,ZFMISC_1:56;
     end;
     let x;
     assume
A8:  x in Support(p)\{b};
     then reconsider x as bag of O;
A9:  x in Support(p) & x<>b by A8,ZFMISC_1:56;
     M.x = 0_(O,R).x by A9,FUNCT_7:32;
     then
A10: M.x = 0.R by POLYNOM1:22;
     (p - M).x = (p+ (-M)).x by POLYNOM1:def 7
     .= (p.x) +((-M).x) by POLYNOM1:15
     .= (p.x) +(-(M.x)) by POLYNOM1:17
     .= p.x by A10;
     then (p - M).x <>0.R by A9,POLYNOM1:def 3;
     hence thesis by A8,A4,POLYNOM1:def 3;
   end;
end;
