
theorem Th3:
for L being add-associative right_zeroed right_complementable
            associative unital right-distributive commutative
            non empty doubleLoopStr
for f being FinSequence of L
for i being Element of NAT st i in dom f
holds Product f = (f/.i) * Product Del(f,i)
proof
let L be add-associative right_zeroed right_complementable
associative unital right-distributive commutative
non empty doubleLoopStr;
let f be FinSequence of L;
let i be Element of NAT;
assume A1: i in dom f;
then i in Seg(len f) by FINSEQ_1:def 3;
then consider ii being Nat such that
A2: ii = i & 1 <= ii & ii <= len f;
reconsider j = i-1 as Element of NAT by A2,INT_1:5;
set g = Del(f,i);
thus Product f = Product( Ins(g,j,f/.i) ) by A1,Th2
         .= Product((g|j)^<*f/.i*>) * Product(g/^j) by GROUP_4:5
         .= (Product(g|j) * f/.i) * Product(g/^j) by GROUP_4:6
         .= f/.i * (Product(g|j) * Product(g/^j)) by GROUP_1:def 3
         .= f/.i * (Product((g|j)^(g/^j))) by GROUP_4:5
         .= f/.i * Product g by RFINSEQ:8;
end;
