reserve x,y,z for Real,
  x3,y3 for Real,
  p for Point of TOP-REAL 3;
reserve p1,p2,p3,p4 for Point of TOP-REAL 3,
  x1,x2,y1,y2,z1,z2 for Real;

theorem Th5:
  p1 + p2 = |[ p1`1 + p2`1, p1`2 + p2`2, p1`3 + p2`3]|
proof
  reconsider p19=p1, p29=p2 as Element of REAL 3 by EUCLID:22;
  len(p19+p29) = 3 by CARD_1:def 7;
  then
A1: dom(p19+p29) = Seg 3 by FINSEQ_1:def 3;
  then 2 in dom(p19+p29) by FINSEQ_1:1;
  then (p19+p29).2 = p19.2 + p29.2 by VALUED_1:def 1;
  then
A2: (p1+p2)`2 = p1`2 + p2`2;
  3 in dom(p19+p29) by A1,FINSEQ_1:1;
  then (p19+p29).3 = p19.3 + p29.3 by VALUED_1:def 1;
  then
A3: (p1+p2)`3 = p1`3 + p2`3;
  1 in dom(p19+p29) by A1,FINSEQ_1:1;
  then (p19+p29).1 = p19.1 + p29.1 by VALUED_1:def 1;
  then (p1+p2)`1 = p1`1 + p2`1;
  hence thesis by A2,A3,Th3;
end;
