reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;

theorem
  for a, b being set holds <*a*> +* (1,b) = <*b*>
proof
  let a, b be set;
A1: dom <*b*> = {1} by FINSEQ_1:2,def 8;
A2: dom <*a*> = {1} by FINSEQ_1:2,def 8;
  then 1 in dom <*a*> by TARSKI:def 1;
  then
A3: <*a*> +* (1,b) = <*a*> +* (1 .--> b) by Def2;
A4: for x being object st x in {1} holds (<*a*> +* (1,b)).x = <*b*>.x
  proof
    let x be object;
    assume x in {1};
    then
A5: x = 1 by TARSKI:def 1;
    hence (<*a*> +* (1,b)).x = b by A3,Th93
      .= <*b*>.x by A5;
  end;
  dom (<*a*> +* (1,b)) = dom <*a*> \/ dom (1 .--> b) by A3,FUNCT_4:def 1
    .= {1} \/ {1} by A2
    .= {1};
  hence thesis by A1,A4;
end;
