reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;
reserve f1, f2 for non empty homogeneous to-naturals NAT*-defined Function,
  e1, e2 for homogeneous to-naturals NAT*-defined Function,
  p for Element of (arity f1+1)-tuples_on NAT;
reserve P for primitive-recursively_closed non empty Subset of HFuncs NAT;

theorem Th82:
  for f being 2-ary len-total to-naturals homogeneous NAT*
  -defined Function holds ((1,2)->(1,?,2) f).<*i,j,k*> = f.<*i,k*>
proof
  let f be 2-ary len-total to-naturals homogeneous NAT*-defined Function;
  reconsider ff=f as quasi_total homogeneous non empty PartFunc of NAT*, NAT
  by Th16;
  reconsider i1 = i, j1 = j,k1 = k as Element of NAT by ORDINAL1:def 12;
  reconsider ijk = <*i1,j1,k1*> as Element of 3-tuples_on NAT
  by FINSEQ_2:104;
A1: arity ff = 2 by Def21;
  then
A2: dom <:<*3 proj 1, 3 proj 3*>:> = dom (3 proj 1) /\ dom (3 proj 3) by Lm8;
A3: dom <:<*3 proj 1, 3 proj 3*>:> = (3-tuples_on NAT) by A1,Lm8;
  dom (ff*<:<*3 proj 1, 3 proj 3*>:>) = 3-tuples_on NAT by A1,Lm8;
  hence ((1,2)->(1,?,2) f).<*i,j,k*> = f.(<:<*3 proj 1, 3 proj 3*>:>.ijk) by
FUNCT_1:12
    .= f.<*(3 proj 1).ijk, (3 proj 3).ijk*> by A2,A3,FINSEQ_3:142
    .= f.<*ijk.1, (3 proj 3).ijk*> by Th37
    .= f.<*ijk.1, ijk.3*> by Th37
    .= f.<*i,k*>;
end;
