reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;
reserve i,j,i1,j1 for Nat;
reserve k for Nat, G for Matrix of D;
reserve x,y,x1,x2,y1,y2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of D;

theorem
  [i,j] in Indices ((n,m)-->a) implies ((n,m)-->a) * (i,j)=a
proof
  reconsider m1=m as Nat;
  set M=(n,m)-->a;
  assume
A1: [i,j] in Indices M;
  then i in dom M by ZFMISC_1:87;
  then i in Seg len M by FINSEQ_1:def 3;
  then
A2: i in Seg n by Def2;
  then
A3: n > 0;
  j in Seg width M by A1,ZFMISC_1:87;
  then j in Seg m by A3,Th23;
  then
A4: (m1|->a).j = a by FUNCOP_1:7;
  M.i=m1|->a by A2,FUNCOP_1:7;
  hence thesis by A1,A4,Def5;
end;
