let I be non empty set ; :: thesis: for i being Element of I

for A being PLS-yielding ManySortedSet of I

for B being Block of (A . i)

for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let i be Element of I; :: thesis: for A being PLS-yielding ManySortedSet of I

for B being Block of (A . i)

for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let A be PLS-yielding ManySortedSet of I; :: thesis: for B being Block of (A . i)

for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let B be Block of (A . i); :: thesis: for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let P be Element of Carrier A; :: thesis: product ({P} +* (i,B)) is Block of (Segre_Product A)

reconsider PP = {P} as ManySortedSubset of Carrier A by PENCIL_1:11;

B in the topology of (A . i) ;

then reconsider B1 = B as Subset of (A . i) ;

then not B1 is trivial by PENCIL_1:4;

then reconsider S = PP +* (i,B1) as non trivial-yielding Segre-like ManySortedSubset of Carrier A by A1, PENCIL_1:9, PENCIL_1:def 20, PENCIL_2:7;

then S . (indx S) = B1 by A2, FUNCT_7:31;

hence product ({P} +* (i,B)) is Block of (Segre_Product A) by A2, PENCIL_1:24; :: thesis: verum

for A being PLS-yielding ManySortedSet of I

for B being Block of (A . i)

for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let i be Element of I; :: thesis: for A being PLS-yielding ManySortedSet of I

for B being Block of (A . i)

for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let A be PLS-yielding ManySortedSet of I; :: thesis: for B being Block of (A . i)

for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let B be Block of (A . i); :: thesis: for P being Element of Carrier A holds product ({P} +* (i,B)) is Block of (Segre_Product A)

let P be Element of Carrier A; :: thesis: product ({P} +* (i,B)) is Block of (Segre_Product A)

reconsider PP = {P} as ManySortedSubset of Carrier A by PENCIL_1:11;

B in the topology of (A . i) ;

then reconsider B1 = B as Subset of (A . i) ;

A1: now :: thesis: for j being Element of I st j <> i holds

({P} +* (i,B1)) . j is 1 -element

2 c= card B
by PENCIL_1:def 6;({P} +* (i,B1)) . j is 1 -element

let j be Element of I; :: thesis: ( j <> i implies ({P} +* (i,B1)) . j is 1 -element )

assume j <> i ; :: thesis: ({P} +* (i,B1)) . j is 1 -element

then ({P} +* (i,B1)) . j = {P} . j by FUNCT_7:32;

hence ({P} +* (i,B1)) . j is 1 -element by Th15; :: thesis: verum

end;assume j <> i ; :: thesis: ({P} +* (i,B1)) . j is 1 -element

then ({P} +* (i,B1)) . j = {P} . j by FUNCT_7:32;

hence ({P} +* (i,B1)) . j is 1 -element by Th15; :: thesis: verum

then not B1 is trivial by PENCIL_1:4;

then reconsider S = PP +* (i,B1) as non trivial-yielding Segre-like ManySortedSubset of Carrier A by A1, PENCIL_1:9, PENCIL_1:def 20, PENCIL_2:7;

A2: now :: thesis: not indx S <> i

dom {P} = I
by PARTFUN1:def 2;assume
indx S <> i
; :: thesis: contradiction

then S . (indx S) is 1 -element by A1;

hence contradiction by PENCIL_1:def 21; :: thesis: verum

end;then S . (indx S) is 1 -element by A1;

hence contradiction by PENCIL_1:def 21; :: thesis: verum

then S . (indx S) = B1 by A2, FUNCT_7:31;

hence product ({P} +* (i,B)) is Block of (Segre_Product A) by A2, PENCIL_1:24; :: thesis: verum