reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem Th3:
  for N being Subgroup of G,H being Subgroup of G,
      x being Element of G st x * N meets carr(H)
   ex y being Element of G st y in x * N & y in H
proof
  let N be Subgroup of G,H being Subgroup of G, x being Element of G;
  assume x * N meets carr(H);
  then consider y be object such that
A1: y in x * N & y in carr(H) by XBOOLE_0:3;
  reconsider y as Element of G by A1;
  y in H by A1,STRUCT_0:def 5;
  hence thesis by A1;
end;
