reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;

theorem
  for S being non empty SubSpace of T,
      t1,t2 being Point of T, s1,s2 being Point of S,
      A,B being Path of t1,t2, C,D being Path of s1,s2 st
  s1,s2 are_connected & t1,t2 are_connected & A = C & B = D &
  Class(EqRel(S,s1,s2),C) = Class(EqRel(S,s1,s2),D) holds
  Class(EqRel(T,t1,t2),A) = Class(EqRel(T,t1,t2),B)
  proof
    let S be non empty SubSpace of T;
    let t1,t2 be Point of T;
    let s1,s2 be Point of S;
    let A,B be Path of t1,t2;
    let C,D be Path of s1,s2 such that
A1: s1,s2 are_connected and
A2: t1,t2 are_connected and
A3: A = C & B = D;
    assume Class(EqRel(S,s1,s2),C) = Class(EqRel(S,s1,s2),D);
    then C,D are_homotopic by A1,TOPALG_1:46;
    then A,B are_homotopic by A1,A2,A3,Th6;
    hence thesis by A2,TOPALG_1:46;
  end;
