
theorem Th26:
  for I being non empty set for A,B being Segre-like non
  trivial-yielding ManySortedSet of I st 2 c= card ((product A) /\ (product B))
  holds indx(A) = indx(B) &
   for i being object st i <> indx(A) holds A.i = B.i
proof
  let I be non empty set;
  let A,B be Segre-like non trivial-yielding ManySortedSet of I;
A1: dom B = I by PARTFUN1:def 2;
  assume 2 c= card ((product A) /\ (product B));
  then consider a,b being object such that
A2: a in (product A) /\ (product B) and
A3: b in (product A) /\ (product B) and
A4: a<>b by Th2;
  b in (product A) by A3,XBOOLE_0:def 4;
  then consider b1 being Function such that
A5: b1=b and
A6: dom b1=dom A and
A7: for o being object st o in dom A holds b1.o in A.o by CARD_3:def 5;
  a in (product A) by A2,XBOOLE_0:def 4;
  then consider a1 being Function such that
A8: a1=a and
A9: dom a1=dom A and
A10: for o being object st o in dom A holds a1.o in A.o by CARD_3:def 5;
  consider o being object such that
A11: o in dom A and
A12: a1.o <> b1.o by A4,A8,A9,A5,A6,FUNCT_1:2;
  reconsider o as Element of I by A11,PARTFUN1:def 2;
  b in (product B) by A3,XBOOLE_0:def 4;
  then
A13: b1.o in B.o by A5,A1,CARD_3:9;
A14: a in (product B) by A2,XBOOLE_0:def 4;
  then a1.o in B.o by A8,A1,CARD_3:9;
  then 2 c= card (B.o) by A12,A13,Th2;
  then
A15: B.o is non trivial by Th4;
  then
A16: o = indx(B) by Def21;
A17: b1.o in A.o by A7,A11;
  a1.o in A.o by A10,A11;
  then 2 c= card (A.o) by A12,A17,Th2;
  then A.o is non trivial by Th4;
  then
A18: o = indx(A) by Def21;
  hence indx(A) = indx(B) by A15,Def21;
  let i be object;
  assume
A19: i <> indx(A);
  per cases;
  suppose
A20: i in I;
    then B.i is 1-element by A18,A16,A19,Th12;
    then
A21: ex y being object st B.i = {y} by ZFMISC_1:131;
    A.i is 1-element by A19,A20,Th12;
    then consider x being object such that
A22: A.i = {x} by ZFMISC_1:131;
    dom B = I by PARTFUN1:def 2;
    then
A23: a1.i in B.i by A8,A14,A20,CARD_3:9;
    dom A = I by PARTFUN1:def 2;
    then a1.i in A.i by A10,A20;
    then a1.i = x by A22,TARSKI:def 1;
    hence thesis by A22,A21,A23,TARSKI:def 1;
  end;
  suppose
A24: not i in I;
    then
A25: not i in dom B by PARTFUN1:def 2;
    not i in dom A by A24,PARTFUN1:def 2;
    hence A.i = {} by FUNCT_1:def 2
      .= B.i by A25,FUNCT_1:def 2;
  end;
end;
