reserve n, k, r, m, i, j for Nat;

theorem Th16:
  for x, y being set, q being FinSubsequence st i < j & q = {[i,x]
  , [j,y]} holds Seq q = <*x,y*>
proof
  let x, y be set, q be FinSubsequence;
  assume that
A1: i < j and
A2: q = {[i,x],[j,y]};
A3: q = (i,j) --> (x,y) by A1,A2,FUNCT_4:67;
  [i,x] in q by A2,TARSKI:def 2;
  then
A4: i in dom q by XTUPLE_0:def 12;
  [j,y] in q by A2,TARSKI:def 2;
  then
A5: j in dom q by XTUPLE_0:def 12;
A6: dom q = {i,j} by A2,RELAT_1:10;
  ex k be Nat st dom q c= Seg k by FINSEQ_1:def 12;
  then i >= 0+1 by A4,FINSEQ_1:1;
  then Seq q = q*<*i,j*> by A1,A6,FINSEQ_3:45
    .= <*q.i,q.j*> by A4,A5,FINSEQ_2:125
    .= <*x,q.j*> by A1,A3,FUNCT_4:63
    .= <*x,y*> by A3,FUNCT_4:63;
  hence thesis;
end;
