 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;

theorem Th7:
  G is idempotent iff for a holds a*a = a
proof
  thus G is idempotent implies for a holds a*a = a
  proof
    assume
A1: op(G).(a,a) = a;
    let a;
    thus thesis by A1;
  end;
  assume
A2: a*a = a;
  let a;
  thus op(G).(a,a) = a*a .= a by A2;
end;
