theorem Th5:
  Suc(f^<*p*>) = p & Ant(f^<*p*>) = f
proof
  set fin = f^<*p*>;
A1: len fin = len f + 1 by FINSEQ_2:16;
  then fin.(len fin) = p by FINSEQ_1:42;
  hence Suc(f^<*p*>) = p by Def2;
  thus Ant(f^<*p*>) = f
  proof
    set fin = f^<*p*>;
    now
      let a be object;
      assume a in f;
      then consider k being Nat such that
A2:   k in dom f and
A3:   a = [k,f.k] by FINSEQ_1:12;
      k in dom fin & f.k = fin.k by A2,FINSEQ_1:def 7,FINSEQ_2:15;
      hence a in fin by A3,FUNCT_1:1;
    end;
    then f c= fin;
    then f = fin|(dom f) by GRFUNC_1:23;
    then f = fin|(Seg len f) by FINSEQ_1:def 3;
    hence thesis by A1,Def1;
  end;
end;
