reserve p, q, x, y for Real,
  n for Nat;
reserve X for non empty TopSpace,
  a, b, c, d, e, f for Point of X,
  T for non empty pathwise_connected TopSpace,
  a1, b1, c1, d1, e1, f1 for Point of T;

theorem Th45:
  a,b are_connected implies for P, Q being Path of a,b holds Q in
  Class(EqRel(X,a,b),P) iff P,Q are_homotopic
proof
  set E = EqRel(X,a,b);
  assume
A1: a,b are_connected;
  let P, Q be Path of a,b;
A2: E is non empty total symmetric transitive by A1,Lm3;
  hereby
    assume Q in Class(E,P);
    then [Q,P] in E by A2,EQREL_1:19;
    hence P,Q are_homotopic by A1,Def3;
  end;
  assume P,Q are_homotopic;
  then [Q,P] in E by A1,Def3;
  hence thesis by A2,EQREL_1:19;
end;
