
theorem NF110:
  for f being FinSequence of NAT, j, b being Nat holds
  b <> j implies (f ^ <* b *>) " {j} = f " {j}
  proof
    let f be FinSequence of NAT, j, b be Nat;

    assume A130: b <> j;

    for z being object holds
    ( z in ((f ^ <* b *>) " {j}) iff z in (f " {j}))
    proof
      let z be object;

      hereby
        assume A200: z in ((f ^ <* b *>) " {j});
        then A210: ( z in dom (f ^ <* b *>) & (f ^ <* b *>) . z in {j} )
        by FUNCT_1:def 7;

        A240: not z in dom (f ^ <* b *>)
        or z in dom f
        or ex n being Nat st
        ( n in dom <* b *> & z = (len f) + n ) by FINSEQ_1:25;

        A260: not ex n being Nat st n in dom <* b *> & z = (len f) + n
        proof
          now given n1 being Nat such that
            B10: n1 in dom <* b *> & z = (len f) + n1;

            dom <* b *> = Seg 1 by FINSEQ_1:def 8;
            then n1 = 1 by B10,FINSEQ_1:2,TARSKI:def 1;
            then B50: (f ^ <* b *>) . z = b by B10,FINSEQ_1:42;

            not b in {j} by A130,TARSKI:def 1;
            hence contradiction by B50,FUNCT_1:def 7,A200;
          end;
          hence thesis;
        end;
        (f ^ <* b *>) . z = f . z
          by A240,FUNCT_1:def 7,A200,A260,FINSEQ_1:def 7;
        hence z in (f " {j}) by A240,A210,A260,FUNCT_1:def 7;
      end;

      assume z in (f " {j});
      then A310: z in dom f & f . z in {j} by FUNCT_1:def 7;

      A315: dom f c= dom (f ^ <* b *>) by FINSEQ_1:26;

      f . z = (f ^ <* b *>) . z by A310,FINSEQ_1:def 7;
      hence z in ((f ^ <* b *>) " {j}) by A315,A310,FUNCT_1:def 7;
    end;
    hence ((f ^ <* b *>) " {j}) = f " {j} by TARSKI:2;
  end;
