let I be non empty set ; :: thesis: for A being PLS-yielding ManySortedSet of I
for X, Y being Subset of () st not X is trivial & X is closed_under_lines & X is strong & not Y is trivial & Y is closed_under_lines & Y is strong & X,Y are_joinable holds
for X1, Y1 being non trivial-yielding Segre-like ManySortedSubset of Carrier A st X = product X1 & Y = product Y1 holds
( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) )

let A be PLS-yielding ManySortedSet of I; :: thesis: for X, Y being Subset of () st not X is trivial & X is closed_under_lines & X is strong & not Y is trivial & Y is closed_under_lines & Y is strong & X,Y are_joinable holds
for X1, Y1 being non trivial-yielding Segre-like ManySortedSubset of Carrier A st X = product X1 & Y = product Y1 holds
( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) )

let X, Y be Subset of (); :: thesis: ( not X is trivial & X is closed_under_lines & X is strong & not Y is trivial & Y is closed_under_lines & Y is strong & X,Y are_joinable implies for X1, Y1 being non trivial-yielding Segre-like ManySortedSubset of Carrier A st X = product X1 & Y = product Y1 holds
( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) ) )

assume that
A1: ( not X is trivial & X is closed_under_lines & X is strong ) and
A2: ( not Y is trivial & Y is closed_under_lines & Y is strong ) and
A3: X,Y are_joinable ; :: thesis: for X1, Y1 being non trivial-yielding Segre-like ManySortedSubset of Carrier A st X = product X1 & Y = product Y1 holds
( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) )

set B = bool the carrier of ();
consider f being FinSequence of bool the carrier of () such that
A4: X = f . 1 and
A5: Y = f . (len f) and
A6: for W being Subset of () st W in rng f holds
( W is closed_under_lines & W is strong ) and
A7: for i being Element of NAT st 1 <= i & i < len f holds
2 c= card ((f . i) /\ (f . (i + 1))) by A3;
len f in dom f by ;
then A8: 1 <= len f by FINSEQ_3:25;
consider B0 being non trivial-yielding Segre-like ManySortedSubset of Carrier A such that
A9: X = product B0 and
for C being Subset of (A . (indx B0)) st C = B0 . (indx B0) holds
( C is strong & C is closed_under_lines ) by ;
let X1, Y1 be non trivial-yielding Segre-like ManySortedSubset of Carrier A; :: thesis: ( X = product X1 & Y = product Y1 implies ( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) ) )

assume that
A10: X = product X1 and
A11: Y = product Y1 ; :: thesis: ( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) )

defpred S1[ Element of NAT ] means for H being non trivial-yielding Segre-like ManySortedSubset of Carrier A st f . \$1 = product H holds
( indx X1 = indx H & ( for i being object st i <> indx X1 holds
X1 . i = H . i ) );
A12: B0 = X1 by ;
A13: for j being Element of NAT st 1 <= j & j < len f & S1[j] holds
S1[j + 1]
proof
let j be Element of NAT ; :: thesis: ( 1 <= j & j < len f & S1[j] implies S1[j + 1] )
assume that
A14: 1 <= j and
A15: j < len f ; :: thesis: ( not S1[j] or S1[j + 1] )
j in dom f by ;
then A16: f . j in rng f by FUNCT_1:3;
A17: 1 <= j + 1 by NAT_1:11;
j + 1 <= len f by ;
then j + 1 in dom f by ;
then f . (j + 1) in rng f by FUNCT_1:3;
then reconsider fj = f . j, fj1 = f . (j + 1) as Subset of () by A16;
A18: card (fj /\ fj1) c= card fj by ;
A19: 2 c= card (fj /\ fj1) by A7, A14, A15;
then 2 c= card fj by A18;
then ( not fj is trivial & fj is closed_under_lines & fj is strong ) by ;
then consider B1 being non trivial-yielding Segre-like ManySortedSubset of Carrier A such that
A20: fj = product B1 and
for C being Subset of (A . (indx B1)) st C = B1 . (indx B1) holds
( C is strong & C is closed_under_lines ) by PENCIL_1:29;
assume A21: S1[j] ; :: thesis: S1[j + 1]
then A22: indx B0 = indx B1 by ;
now :: thesis: for H being non trivial-yielding Segre-like ManySortedSubset of Carrier A st f . (j + 1) = product H holds
( indx X1 = indx H & ( for i being object st i <> indx X1 holds
X1 . i = H . i ) )
let H be non trivial-yielding Segre-like ManySortedSubset of Carrier A; :: thesis: ( f . (j + 1) = product H implies ( indx X1 = indx H & ( for i being object st i <> indx X1 holds
X1 . i = H . i ) ) )

assume A23: f . (j + 1) = product H ; :: thesis: ( indx X1 = indx H & ( for i being object st i <> indx X1 holds
X1 . i = H . i ) )

hence indx X1 = indx H by ; :: thesis: for i being object st i <> indx X1 holds
X1 . i = H . i

thus for i being object st i <> indx X1 holds
X1 . i = H . i :: thesis: verum
proof
let i be object ; :: thesis: ( i <> indx X1 implies X1 . i = H . i )
assume A24: i <> indx X1 ; :: thesis: X1 . i = H . i
then A25: i <> indx B1 by ;
thus X1 . i = B1 . i by
.= H . i by ; :: thesis: verum
end;
end;
hence S1[j + 1] ; :: thesis: verum
end;
A26: S1 by A10, A4, PUA2MSS1:2;
for i being Element of NAT st 1 <= i & i <= len f holds
S1[i] from hence ( indx X1 = indx Y1 & ( for i being object st i <> indx X1 holds
X1 . i = Y1 . i ) ) by A11, A5, A8; :: thesis: verum