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 Th18:
  a |^ b = 1_G implies a = 1_G
proof
  assume a |^ b = 1_G;
  then b" = b" * a by GROUP_1:12;
  hence thesis by GROUP_1:7;
end;
