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;

theorem Th46:
  apply(<*f*>,x) = <*x,f.x*>
proof
A1: <*x*>.(0+1) = x;
A2: apply({},x) = <*x*> & len {} = 0 by Th39;
  thus apply(<*f*>,x) = apply({}^<*f*>,x) by FINSEQ_1:34
    .= <*x*>^<*f.x*> by A2,A1,Th41
    .= <*x,f.x*> by FINSEQ_1:def 9;
end;
