theorem Th47:
  for x being set holds x in the carrier of pi_1(X,a) iff ex P
  being Loop of a st x = Class(EqRel(X,a),P)
proof
  let x be set;
A1: the carrier of pi_1(X,a) = Class EqRel (X,a) by Def5;
  hereby
    assume x in the carrier of pi_1(X,a);
    then consider P being Element of Loops(a) such that
A2: x = Class(EqRel(X,a),P) by A1,EQREL_1:36;
    reconsider P as Loop of a by Def1;
    take P;
    thus x = Class(EqRel(X,a),P) by A2;
  end;
  given P being Loop of a such that
A3: x = Class(EqRel(X,a),P);
  P in Loops(a) by Def1;
  hence thesis by A1,A3,EQREL_1:def 3;
end;
