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

theorem Th10:
  for S being non empty Subset of T
  st n >= 2 & S = ([#]T) \ {t} & TOP-REAL n, T are_homeomorphic
  holds T | S is pathwise_connected
  proof
    let S be non empty Subset of T;
    assume
A1: n >= 2 & S = ([#]T) \ {t} & TOP-REAL n, T are_homeomorphic;
    then consider f be Function of T, TOP-REAL n such that
A2: f is being_homeomorphism by T_0TOPSP:def 1;
    reconsider p = f.t as Point of TOP-REAL n;
    reconsider SN = ([#]TOP-REAL n) \ {p}
    as non empty Subset of TOP-REAL n by A1,RAMSEY_1:4;
A3: (TOP-REAL n) | SN is pathwise_connected by A1,Th9;
A4:dom f = [#]T & rng f = [#]TOP-REAL n by A2,TOPS_2:58;
    then
A5: f"([#]TOP-REAL n) = [#]T by RELAT_1:134;
    consider x be object such that
A6:f"{p} = {x} by A4,A2,FUNCT_1:74;
A7: x in f"{p} by A6,TARSKI:def 1;
    then x in dom f & f.x in {p} by FUNCT_1:def 7;
    then p = f.x by TARSKI:def 1;
    then x = t by A2,A7,A4,FUNCT_1:def 4;
    then
A8: f"SN = S by A1,A5,A6,FUNCT_1:69;
A9: dom(SN |` f) = f"SN by MFOLD_2:1 .= [#](T | f"SN) by PRE_TOPC:def 5;
    rng(SN |` f) c= SN;
    then rng(SN |` f) c= [#]((TOP-REAL n) | SN) by PRE_TOPC:def 5;
    then reconsider g = SN|`f as Function of T | f"SN, (TOP-REAL n) | SN
    by A9,FUNCT_2:2;
    g is being_homeomorphism by A2,MFOLD_2:4;
    then (TOP-REAL n) | SN, T | S are_homeomorphic by A8,T_0TOPSP:def 1;
    hence T | S is pathwise_connected by A3,TOPALG_3:9;
  end;
