
theorem
for R being non degenerated comRing
for M being non empty Subset of R
for o being object
holds o in M-Ideal iff
      ex P being non empty finite Subset of R,
         L being LinearCombination of P st P c= M & o = Sum L
proof
let R be non degenerated comRing, M be non empty Subset of R, o be object;
A: now assume o in M-Ideal; then
   consider P being non empty finite Subset of R,
            L being LeftLinearCombination of P such that
  A1: P c= M & o = Sum L by ideal1;
  L is LinearCombination of P by IDEAL_1:25;
  hence ex P being non empty finite Subset of R,
        L being LinearCombination of P st P c= M & o = Sum L by A1;
  end;
now assume ex P being non empty finite Subset of R,
              L being LinearCombination of P st P c= M & o = Sum L; then
  consider P being non empty finite Subset of R,
           L being LinearCombination of P such that
  A2: P c= M & o = Sum L;
  L is LeftLinearCombination of P by IDEAL_1:31;
  hence o in M-Ideal by A2,ideal1;
  end;
hence thesis by A;
end;
