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 Th25:
  a |^ b |^ b" = a & a |^ b" |^ b = a
proof
  thus a |^ b |^ b" = a |^ (b * b") by Th24
    .= a |^ 1_G by GROUP_1:def 5
    .= a by Th19;
  thus a |^ b" |^ b = a |^ (b" * b) by Th24
    .= a |^ 1_G by GROUP_1:def 5
    .= a by Th19;
end;
