reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;
reserve p,p1,p2,p3 for Point of TOP-REAL n,
  x,x1,x2,y,y1,y2 for Real;
reserve p,p1,p2 for Point of TOP-REAL 2;

theorem Th27:
  p1 + p2 = |[ p1`1 + p2`1, p1`2 + p2`2]|
proof
  the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by Def8;
  then reconsider p19=p1, p29=p2 as Element of REAL 2;
  len(p19+p29) = 2 by CARD_1:def 7;
  then
A1: dom(p19+p29) = Seg 2 by FINSEQ_1:def 3;
  2 in {1,2} by TARSKI:def 2;
  then
A2: (p1+p2)`2 = p1`2 + p2`2 by A1,FINSEQ_1:2,VALUED_1:def 1;
  1 in {1,2} by TARSKI:def 2;
  then (p1+p2)`1 = p1`1 + p2`1 by A1,FINSEQ_1:2,VALUED_1:def 1;
  hence thesis by A2,Th25;
end;
