reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;

theorem Th19:
  for i,j be Element of NAT
  ex f be Function of[:TOP-REAL i,TOP-REAL j:],TOP-REAL(i+j) st
    f is being_homeomorphism &
    for fi be Element of TOP-REAL i,fj be Element of TOP-REAL j
    holds f.(fi,fj) = fi^fj
 proof
  let i,j be Element of NAT;
  set TRi=TOP-REAL i,TRj=TOP-REAL j,TRij=TOP-REAL(i+j);
  set tri=the TopStruct of TRi,trj=the TopStruct of TRj,trij=the TopStruct of
TRij;
  A1: trj=TopSpaceMetr Euclid j by EUCLID:def 8;
  trij=TopSpaceMetr Euclid(i+j) & tri=TopSpaceMetr Euclid i by EUCLID:def 8;
  then consider f be Function of[:tri,trj:],trij such that
   A2: f is being_homeomorphism and
   A3: for fi,fj be FinSequence st[fi,fj] in dom f holds f.(fi,fj)=fi^fj by A1,
TOPREAL7:26;
  rng f=[#]trij by A2,TOPS_2:60;
  then A4: rng f=[#]TRij;
  A5: [#][:TRi,TRj:]=[:[#]TRi,[#]TRj:] by BORSUK_1:def 2;
  A6: [:TRi,TRj:]=[:TRi,TRj:]|([#][:TRi,TRj:]) by TSEP_1:3
   .=[:TRi|([#]TRi),TRj|([#]TRj):] by A5,BORSUK_3:22
   .=[:tri,TRj|([#]TRj):] by TSEP_1:93
   .=[:tri,trj:] by TSEP_1:93;
  then reconsider F=f as Function of[:TRi,TRj:],TRij;
  A7: now let P be Subset of[:TRi,TRj:];
   thus F.:(Cl P)=Cl(f.:P) by A2,A6,TOPS_2:60
    .=Cl(F.:P) by TOPS_3:80;
  end;
  take F;
  A8: F is one-to-one by A2,TOPS_2:60;
  dom F=[#][:TRi,TRj:] by A2,A6,TOPS_2:60;
  hence F is being_homeomorphism by A4,A7,A8,TOPS_2:60;
  let fi be Element of TOP-REAL i,fj be Element of TOP-REAL j;
  dom F=[:[#]TRi,[#]TRj:] by A5,FUNCT_2:def 1;
  then [fi,fj] in dom F by ZFMISC_1:87;
  hence thesis by A3;
 end;
