theorem Th52:
  x0,x1 are_connected implies pi_1-iso(P) is onto
proof
  assume
A1: x0,x1 are_connected;
  set f = pi_1-iso(P);
  thus rng f c= the carrier of pi_1(X,x0);
  let y be object;
  assume y in the carrier of pi_1(X,x0);
  then consider Y being Loop of x0 such that
A2: y = Class(EqRel(X,x0),Y) by Th47;
A3: P+(-P+Y+P)+-P, Y are_homotopic by A1,Th41;
  set Z = Class(EqRel(X,x1),-P+Y+P);
  dom f = the carrier of pi_1(X,x1) by FUNCT_2:def 1;
  then
A4: Z in dom f by Th47;
  f.Z = Class(EqRel(X,x0),P+(-P+Y+P)+-P) by A1,Def6
    .= y by A2,A3,Th46;
  hence thesis by A4,FUNCT_1:def 3;
end;
