reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;

theorem Th12:
  for G being non trivial Group
  for H being proper Subgroup of G
  for K being Subgroup of G
  st H is Subgroup of K & the multMagma of H <> the multMagma of K
  holds K is non trivial Subgroup of G
proof
  let G be non trivial Group;
  let H be proper Subgroup of G;
  let K be Subgroup of G;
  assume A1: H is Subgroup of K;
  assume A2: the multMagma of H <> the multMagma of K;
  assume B1: not K is non trivial Subgroup of G;
  then H is trivial Subgroup of G by A1,Th9;
  hence contradiction by A2,B1,Th8;
end;
