reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  for H being strict Subgroup of G holds H |^ a = G implies H = G
proof
  let H be strict Subgroup of G;
  assume
A1: H |^ a = G;
  now
    let g;
    assume
A2: not g in H;
    now
      assume g |^ a in H |^ a;
      then ex h st g |^ a = h |^ a & h in H by Th58;
      hence contradiction by A2,Th16;
    end;
    hence contradiction by A1;
  end;
  hence thesis by A1,GROUP_2:62;
end;
