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

theorem Th25:
  for L be non empty ZeroStr
    for b be bag of X holds Monom(0.L,b) = 0_(X,L)
proof
  let L be non empty ZeroStr,b be bag of X;
A1: dom 0_(X,L) = Bags X by FUNCT_2:def 1;
  x in Bags X implies Monom(0.L,b).x = 0_(X,L).x
  proof
    assume
A2:  x in Bags X;
    per cases;
    suppose x=b;
      hence Monom(0.L,b).x = 0.L by A1,A2,FUNCT_7:31
      .= 0_(X,L).x by A2,POLYNOM1:22;
    end;
    suppose x<>b;
      hence Monom(0.L,b).x = 0_(X,L).x by FUNCT_7:32;
    end;
  end;
  hence thesis;
end;
