reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;

theorem Th36:
  p = <%x,y,z%> iff len p = 3 & p.0 = x & p.1 = y & p.2 = z
proof
  thus p = <%x,y,z%> implies len p = 3 & p.0 = x & p.1 = y & p.2 = z
  proof
A2: 0 in dom <%x%> by TARSKI:def 1;
A3: 0 in dom <%z%> by TARSKI:def 1;
    assume
A4: p =<%x,y,z%>;
    hence len p =len <%x,y%> + len <%z%> by Def3
      .=2 + len <%z%> by Th35
      .=2+1 by Th30
      .=3;
    thus p.0 = (<%x%>^<%y,z%>).0 by A4,Th25
      .=<%x%>.0 by A2,Def3
      .=x;
    1 in Segm(1+1) & len <%x,y%> = 2 by Th35,NAT_1:45;
    hence p.1 =<%x,y%>.1 by A4,Def3
      .=y;
    thus p.2 =(<%x,y%>^<%z%>).(len (<%x,y%>) + 0) by A4,Th35
      .= <%z%>.0 by A3,Def3
      .= z;
  end;
  assume that
A5: len p = 3 and
A6: p.0 = x and
A7: p.1 = y and
A8: p.2 = z;
A9: for k st k in dom <%x,y%> holds p.k=<%x,y%>.k
  proof
A10: len <%x,y%> = 2 by Th35;
    let k such that
A11: k in dom <%x,y%>;
A12: k=1 implies p.k=<%x,y%>.k by A7;
    k=0 implies p.k=<%x,y%>.k by A6;
    hence thesis by A11,A10,A12,CARD_1:50,TARSKI:def 2;
  end;
A13: for k st k in dom <%z%> holds p.( (len <%x,y%>) + k) = <%z%>.k
  proof
    let k;
    assume k in dom <%z%>;
    then
A14: k = 0 by TARSKI:def 1;
    hence p.( (len <%x,y%>) + k) = p.(2+0) by Th35
      .=<%z%>.k by A8,A14;
  end;
  dom p = (2+1) by A5
    .= ((len <%x,y%>) + 1) by Th35
    .= ((len <%x,y%>) + len <%z%>) by Th30;
  hence thesis by A9,A13,Def3;
end;
