
theorem
  for L being non empty multLoopStr, P,Q being non empty Subset of L, A
  being LeftLinearCombination of P holds P c= Q implies A is
  LeftLinearCombination of Q
proof
  let L be non empty multLoopStr, P,Q be non empty Subset of L, A be
  LeftLinearCombination of P;
  assume
A1: P c= Q;
  now
    let i be set;
    assume i in dom A;
    then consider u being Element of L, a being Element of P such that
A2: A/.i = u*a by IDEAL_1:def 9;
    a in P;
    hence ex u being Element of L, a being Element of Q st A/.i = u*a by A1,A2;
  end;
  hence thesis by IDEAL_1:def 9;
end;
