set S = SgmX ((RelIncl n),(support b));
set l = len (SgmX ((RelIncl n),(support b)));
defpred S₁[ Element of NAT , Element of L] means $2 = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. $1),((b * (SgmX ((RelIncl n),(support b)))) /. $1));
A1: for k being Element of NAT st k in Seg (len (SgmX ((RelIncl n),(support b)))) holds
ex x being Element of L st S₁[k,x] ;
consider p being FinSequence of the carrier of L such that
A2: ( dom p = Seg (len (SgmX ((RelIncl n),(support b)))) & ( for k being Element of NAT st k in Seg (len (SgmX ((RelIncl n),(support b)))) holds
S₁[k,p /. k] ) ) from RECDEF_1:sch 17(A1);
take Product p ; :: thesis: ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & Product p = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) )
A3: len p = len (SgmX ((RelIncl n),(support b))) by A2, FINSEQ_1:def 3;

now

let m be Element of NAT ; :: thesis: ( 1 <= m & m <= len p implies p /. m = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. m),((b * (SgmX ((RelIncl n),(support b)))) /. m)) )
assume that
A4: 1 <= m and
A5: m <= len p ; :: thesis: p /. m = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. m),((b * (SgmX ((RelIncl n),(support b)))) /. m))
m in Seg (len (SgmX ((RelIncl n),(support b)))) by A3, A4, A5, FINSEQ_1:1;
hence p /. m = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. m),((b * (SgmX ((RelIncl n),(support b)))) /. m)) by A2; :: thesis: verum

end;

hence ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & Product p = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) by A3; :: thesis: verum