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;

theorem Th11:
  for S be Simplex of BCS Kv
  for F be c=-linear finite finite-membered Subset-Family of V st
    S = (center_of_mass V).:F
  for a1,a2 be VECTOR of V st a1 in S & a2 in S
  ex b1,b2 be VECTOR of V,r be Real st
     b1 in Vertices BCS Complex_of{union F} &
     b2 in Vertices BCS Complex_of{union F} &
     a1-a2 = r*(b1-b2) & 0 <= r & r <= (card union F-1)/card union F
 proof
  let S be Simplex of BCS Kv;
  set cM=center_of_mass V;
  let F be c=-linear finite finite-membered Subset-Family of V such that
   A1: S=cM.:F;
  let a1,a2 be VECTOR of V such that
   A2: a1 in S and
   A3: a2 in S;
  consider A1 be object such that
   A4: A1 in dom cM and
   A5: A1 in F and
   A6: cM.A1=a1 by A1,A2,FUNCT_1:def 6;
  consider A2 be object such that
   A7: A2 in dom cM and
   A8: A2 in F and
   A9: cM.A2=a2 by A1,A3,FUNCT_1:def 6;
  reconsider A1,A2 as set by TARSKI:1;
  A10: A1,A2 are_c=-comparable by A5,A8,ORDINAL1:def 8;
  set uF=union F;
  set CuF=Complex_of{uF};
  A11: for a1,a2 be VECTOR of V for A1,A2 be set st A1 c=A2 & A1 in dom cM & A1
in F & cM.A1=a1 & A1 in dom cM & A2 in F & cM.A2=a2 ex b1,b2 be VECTOR of V,r
be Real st b1 in Vertices BCS CuF & b2 in Vertices BCS CuF & a1-a2=r*(b1-b2) &
0<=r & r<=(card uF-1)/card uF
  proof
   let a1,a2 be VECTOR of V;
   A12: the topology of CuF=bool uF by SIMPLEX0:4;
   let A1,A2 be set such that
    A13: A1 c=A2 and
    A14: A1 in dom cM and
    A15: A1 in F and
    A16: cM.A1=a1 and
    A1 in dom cM and
    A17: A2 in F and
    A18: cM.A2=a2;
   A19: A1 c=uF by A15,ZFMISC_1:74;
   A20: A1<>{} by A14,ORDERS_1:1;
   then A21: uF is non empty by A19;
   A22: A2 c=uF by A17,ZFMISC_1:74;
   reconsider A1,A2 as non empty finite Subset of V by A13,A15,A17,A20;
   set A21=A2\A1;
   reconsider AA1={A1},AA2={A21} as Subset-Family of CuF;
   {A1}c=bool uF by A19,ZFMISC_1:31;
   then A23: AA1 is c=-linear finite simplex-like by A12,Lm2,SIMPLEX0:14;
   A21 c=A2 by XBOOLE_1:36;
   then A21 c=uF by A22;
   then {A21}c=bool uF by ZFMISC_1:31;
   then A24: AA2 is c=-linear finite simplex-like by A12,Lm2,SIMPLEX0:14;
   A25: |.CuF.|c=[#]V;
   [#]CuF=[#]V;
   then A26: BCS CuF=subdivision(cM,CuF) by A25,SIMPLEX1:def 5;
   A27: [#]subdivision(cM,CuF)=[#]CuF by SIMPLEX0:def 20;
   then reconsider aa1={a1} as Subset of BCS CuF by A25,SIMPLEX1:def 5;
   A28: a1 in aa1 by TARSKI:def 1;
   then reconsider d1=a1 as Element of BCS CuF;
   A29: dom cM=BOOL the carrier of V by FUNCT_2:def 1;
   cM.:AA1=Im(cM,A1) by RELAT_1:def 16
    .={a1} by A14,A16,FUNCT_1:59;
   then aa1 is simplex-like by A23,A26,SIMPLEX0:def 20;
   then A30: d1 is vertex-like by A28;
   per cases;
   suppose A31: A1=A2;
    reconsider Z=0 as Real;
    take b1=a1,b2=a1,Z;
    card uF>=1 by A21,NAT_1:14;
    then A32: card uF-1>=1-1 by XREAL_1:6;
    a1-a2=0.V by A16,A18,A31,RLVECT_1:5;
    hence thesis by A30,A32,RLVECT_1:10,SIMPLEX0:def 4;
   end;
   suppose A1<>A2;
    then not A2 c=A1 by A13,XBOOLE_0:def 10;
    then reconsider A21 as non empty finite Subset of V by XBOOLE_1:37;
    A33: A21 in dom cM by A29,ORDERS_1:2;
    then cM.A21 in rng cM by FUNCT_1:def 3;
    then reconsider a21=cM.A21 as VECTOR of V;
    reconsider aa2={a21} as Subset of BCS CuF by A25,A27,SIMPLEX1:def 5;
    A34: a21 in aa2 by TARSKI:def 1;
    then reconsider d2=a21 as Element of BCS CuF;
    cM.:AA2=Im(cM,A21) by RELAT_1:def 16
     .={a21} by A33,FUNCT_1:59;
    then aa2 is simplex-like by A24,A26,SIMPLEX0:def 20;
    then A35: d2 is vertex-like by A34;
    set r1=1/card A1,r2=1/card A2,r21=1/card A21,r=card A21/card A2;
    reconsider r1,r2,r21,r as Real;
    take a1,a21,r;
    A36: r*r21=(card A21*1)/(card A21*card A2) by XCMPLX_1:76
     .=r2 by XCMPLX_1:91;
    consider L1 be Linear_Combination of A1 such that
     A37: Sum L1=r1*Sum A1 and
     sum L1=r1*card A1 and
     A38: L1=(ZeroLC V)+*(A1-->r1) by RLAFFIN2:15;
    A39: Carrier(L1)c=A1 by RLVECT_2:def 6;
    A40: card A21=1 *card A2-1 *card A1 by A13,CARD_2:44;
    then A41: r1-r2=(card A21*1)/(card A2*card A1) by XCMPLX_1:130
     .=r*r1 by XCMPLX_1:76;
    consider L21 be Linear_Combination of A21 such that
     A42: Sum L21=r21*Sum A21 and
     sum L21=r21*card A21 and
     A43: L21=(ZeroLC V)+*(A21-->r21) by RLAFFIN2:15;
    A44: Carrier(L21)c=A21 by RLVECT_2:def 6;
    consider L2 be Linear_Combination of A2 such that
     A45: Sum L2=r2*Sum A2 and
     sum L2=r2*card A2 and
     A46: L2=(ZeroLC V)+*(A2-->r2) by RLAFFIN2:15;
    A47: Carrier(L2)c=A2 by RLVECT_2:def 6;
    for v be Element of V holds(L1-L2).v=(r*(L1-L21)).v
    proof
     let v be Element of V;
     A48: dom(A21-->r21)=A21 by FUNCOP_1:13;
     A49: (L1-L2).v=L1.v-L2.v by RLVECT_2:54;
     A50: dom(A1-->r1)=A1 by FUNCOP_1:13;
     A51: dom(A2-->r2)=A2 by FUNCOP_1:13;
     (r*(L1-L21)).v=r*((L1-L21).v) by RLVECT_2:def 11;
     then A52: (r*(L1-L21)).v=r*(L1.v-L21.v) by RLVECT_2:54;
     per cases;
     suppose A53: v in A1;
      then not v in Carrier(L21) by A44,XBOOLE_0:def 5;
      then A54: L21.v=0 by RLVECT_2:19;
      A55: (A2-->r2).v=r2 & (A1-->r1).v=r1 by A13,A53,FUNCOP_1:7;
      L1.v=(A1-->r1).v by A38,A50,A53,FUNCT_4:13;
      hence thesis by A13,A41,A46,A49,A51,A52,A53,A54,A55,FUNCT_4:13;
     end;
     suppose A56: v in A2 & not v in A1;
      then not v in Carrier L1 by A39;
      then A57: L1.v=0 by RLVECT_2:19;
      (A2-->r2).v=r2 by A56,FUNCOP_1:7;
      then A58: (L1-L2).v=-r2 by A46,A49,A51,A56,A57,FUNCT_4:13;
      A59: v in A21 by A56,XBOOLE_0:def 5;
      then (A21-->r21).v=r21 by FUNCOP_1:7;
      then (r*(L1-L21)).v=r*(-r21) by A43,A48,A52,A57,A59,FUNCT_4:13;
      hence thesis by A36,A58;
     end;
     suppose A60: not v in A1 & not v in A2;
      then not v in Carrier(L1) by A39;
      then A61: L1.v=0 by RLVECT_2:19;
      not v in Carrier(L21) by A44,A60,XBOOLE_0:def 5;
      then A62: L21.v=0 by RLVECT_2:19;
      not v in Carrier(L2) by A47,A60;
      hence thesis by A49,A52,A62,A61,RLVECT_2:19;
     end;
    end;
    then A63: L1-L2=r*(L1-L21) by RLVECT_2:def 9;
    card A1>=1 & card A2<=card uF by A17,NAT_1:14,43,ZFMISC_1:74;
    then card A1*card uF>=1 *card A2 by XREAL_1:66;
    then card A2*card uF-card A1*card uF<=card uF*card A2-card A2 by XREAL_1:13
;
    then A64: card A21*card uF<=(card uF-1)*card A2 by A40;
    A65: Sum L21=a21 by A42,RLAFFIN2:def 2;
    A66: Sum L1=a1 by A16,A37,RLAFFIN2:def 2;
    Sum L2=a2 by A18,A45,RLAFFIN2:def 2;
    then a1-a2=Sum(r*(L1-L21)) by A63,A66,RLVECT_3:4
     .=r*Sum(L1-L21) by RLVECT_3:2
     .=r*(a1-a21) by A65,A66,RLVECT_3:4;
    hence thesis by A21,A30,A35,A64,SIMPLEX0:def 4,XREAL_1:102;
   end;
  end;
  per cases by A10,XBOOLE_0:def 9;
  suppose A1 c=A2;
   hence thesis by A4,A5,A6,A8,A9,A11;
  end;
  suppose A2 c=A1;
   then consider b1,b2 be VECTOR of V,r be Real such that
    A67: b1 in Vertices BCS CuF & b2 in Vertices BCS CuF and
    A68: a2-a1=r*(b1-b2) and
    A69: 0<=r & r<=(card uF-1)/card uF by A5,A6,A7,A8,A9,A11;
   take b2,b1,r;
   a1-a2=-(r*(b1-b2)) by A68,RLVECT_1:33
    .=r*(-(b1-b2)) by RLVECT_1:25
    .=r*(b2-b1) by RLVECT_1:33;
   hence thesis by A67,A69;
  end;
 end;
