
theorem FStoMAT1:
for S be non empty cap-closed set, F1,F2 be FinSequence of S holds
 ex Mx be Matrix of len F1,len F2,S st
  for i,j be Nat st [i,j] in Indices Mx holds Mx*(i,j) = F1.i /\ F2.j
proof
   let S be non empty cap-closed set;
   let F1,F2 be FinSequence of S;

   defpred P[Nat,Nat,set] means $3 = F1.$1 /\ F2.$2;

A2:for i,j be Nat st [i,j] in [:Seg len F1, Seg len F2:]
    ex K be Element of S st P[i,j,K]
   proof
    let i,j be Nat;
    assume [i,j] in [:Seg len F1,Seg len F2:]; then
    i in Seg len F1 & j in Seg len F2 by ZFMISC_1:87; then
    i in dom F1 & j in dom F2 by FINSEQ_1:def 3; then
    F1.i in rng F1 & F2.j in rng F2 by FUNCT_1:3; then
    F1.i /\ F2.j in S by FINSUB_1:def 2;
    hence thesis;
   end;

   consider Mx be Matrix of len F1,len F2,S such that
A3: for i,j be Nat st
       [i,j] in Indices Mx holds P[i,j,Mx*(i,j)] from MATRIX_0:sch 2(A2);
   take Mx;
   thus thesis by A3;
end;
