
theorem Th4:
  for FT1,FT2 being non empty RelStr, h being Function of FT1, FT2,
n being Nat, x being Element of FT1, y being Element of FT2 st h is
being_homeomorphism & y=h.x holds for z being Element of FT1 holds z in U_FT(x,
  n) iff h.z in U_FT(y,n)
proof
  let FT1,FT2 be non empty RelStr, h be Function of FT1, FT2, n be Nat,
      x be Element of FT1,y be Element of FT2;
  assume that
A1: h is being_homeomorphism and
A2: y=h.x;
A3: h is one-to-one onto by A1;
  let z be Element of FT1;
  x in the carrier of FT1;
  then
A4: x in dom h by FUNCT_2:def 1;
  z in the carrier of FT1;
  then
A5: z in dom h by FUNCT_2:def 1;
A6: now
    defpred P[Nat] means for w being Element of FT2 holds w in U_FT
    (y,$1) implies h".w in U_FT(x,$1);
    assume
A7: h.z in U_FT(y,n);
    consider g being Function of FT2, FT1 such that
A8: g=h" and
A9: g is being_homeomorphism by A1,Th3;
A10: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A11:  P[k];
      for w being Element of FT2 holds w in U_FT(y,k+1) implies h".w in
      U_FT(x,k+1)
      proof
        let w be Element of FT2;
A12:    U_FT(y,k+1)=(U_FT(y,k))^f by FINTOPO3:48;
        assume w in U_FT(y,k+1);
        then consider x3 being Element of FT2 such that
A13:    x3=w and
A14:    ex y3 being Element of FT2 st y3 in U_FT(y,k) & x3 in U_FT y3 by A12;
        consider y2 being Element of FT2 such that
A15:    y2 in U_FT(y,k) and
A16:    x3 in U_FT y2 by A14;
        reconsider q=g.y2, p=g.x3 as Element of FT1;
A17:    for w2 being Element of FT2 holds w2 in U_FT(y2,0) implies h".w2
        in U_FT(q,0)
        proof
          let w2 be Element of FT2;
          w2 in the carrier of FT2;
          then
A18:      w2 in dom g by FUNCT_2:def 1;
A19:      h".:U_FT(y2)=Class(the InternalRel of FT1,h".y2) by A8,A9;
          hereby
            assume w2 in U_FT(y2,0);
            then w2 in U_FT y2 by FINTOPO3:47;
            then h".w2 in U_FT q by A8,A19,A18,FUNCT_1:def 6;
            hence h".w2 in U_FT(q,0) by FINTOPO3:47;
          end;
        end;
        x3 in U_FT(y2,0) by A16,FINTOPO3:47;
        then p in U_FT(q,0) by A8,A17;
        then
A20:    p in U_FT q by FINTOPO3:47;
        q in U_FT(x,k) by A8,A11,A15;
        then p in (U_FT(x,k))^f by A20;
        hence thesis by A8,A13,FINTOPO3:48;
      end;
      hence thesis;
    end;
A21: g.y=x by A2,A4,A3,A8,FUNCT_1:34;
    for w being Element of FT2 holds w in U_FT(y,0) implies h".w in U_FT( x,0)
    proof
      let w be Element of FT2;
      w in the carrier of FT2;
      then
A22:  w in dom g by FUNCT_2:def 1;
A23:  g.:U_FT(y)=Class(the InternalRel of FT1,g.y) by A9;
      hereby
        assume w in U_FT(y,0);
        then w in U_FT y by FINTOPO3:47;
        then g.w in U_FT x by A21,A23,A22,FUNCT_1:def 6;
        hence h".w in U_FT(x,0) by A8,FINTOPO3:47;
      end;
    end;
    then
A24: P[0];
    for k being Nat holds P[k] from NAT_1:sch 2(A24,A10);
    then h".(h.z) in U_FT(x,n) by A7;
    hence z in U_FT(x,n) by A3,A5,FUNCT_1:34;
  end;
  now
    defpred P[Nat] means for w being Element of FT1 holds w in U_FT
    (x,$1) implies h.w in U_FT(y,$1);
    assume
A25: z in U_FT(x,n);
A26: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A27:  P[k];
      for w being Element of FT1 holds w in U_FT(x,k+1) implies h.w in
      U_FT(y,k+1)
      proof
        let w be Element of FT1;
A28:    U_FT(x,k+1)=(U_FT(x,k))^f by FINTOPO3:48;
        assume w in U_FT(x,k+1);
        then consider x3 being Element of FT1 such that
A29:    x3=w and
A30:    ex y3 being Element of FT1 st y3 in U_FT(x,k) & x3 in U_FT y3 by A28;
        consider y2 being Element of FT1 such that
A31:    y2 in U_FT(x,k) and
A32:    x3 in U_FT y2 by A30;
        reconsider q=h.y2, p=h.x3 as Element of FT2;
A33:    for w2 being Element of FT1 holds w2 in U_FT(y2,0) implies h.w2
        in U_FT(q,0)
        proof
          let w2 be Element of FT1;
          w2 in the carrier of FT1;
          then
A34:      w2 in dom h by FUNCT_2:def 1;
A35:      h.:U_FT(y2)=Class(the InternalRel of FT2,h.y2) by A1;
          hereby
            assume w2 in U_FT(y2,0);
            then w2 in U_FT y2 by FINTOPO3:47;
            then h.w2 in U_FT q by A35,A34,FUNCT_1:def 6;
            hence h.w2 in U_FT(q,0) by FINTOPO3:47;
          end;
        end;
        x3 in U_FT(y2,0) by A32,FINTOPO3:47;
        then p in U_FT(q,0) by A33;
        then
A36:    p in U_FT q by FINTOPO3:47;
        q in U_FT(y,k) by A27,A31;
        then p in (U_FT(y,k))^f by A36;
        hence thesis by A29,FINTOPO3:48;
      end;
      hence thesis;
    end;
    for w being Element of FT1 holds w in U_FT(x,0) implies h.w in U_FT(y ,0)
    proof
      let w be Element of FT1;
      w in the carrier of FT1;
      then
A37:  w in dom h by FUNCT_2:def 1;
A38:  h.:U_FT(x)=Class(the InternalRel of FT2,h.x) by A1;
      hereby
        assume w in U_FT(x,0);
        then w in U_FT x by FINTOPO3:47;
        then h.w in U_FT y by A2,A38,A37,FUNCT_1:def 6;
        hence h.w in U_FT(y,0) by FINTOPO3:47;
      end;
    end;
    then
A39: P[0];
    for k being Nat holds P[k] from NAT_1:sch 2(A39,A26);
    hence h.z in U_FT(y,n) by A25;
  end;
  hence thesis by A6;
end;
