reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem
  p,q // r,s implies p,q // sum(p,r,o),sum(q,s,o)
proof
  assume
A1: p,q // r,s;
  now
    consider t such that
A2: congr o,q,r,t by Th63;
    assume that
A3: p<>q and
A4: r<>s;
    congr o,q,s,sum(q,s,o) by Def5;
    then congr r,t,s,sum(q,s,o) by A2,Th65;
    then
A5: congr r,s,t,sum(q,s,o) by Th69;
    then
A6: t<>sum(q,s,o) by A4,Th55;
    r,s // t,sum(q,s,o) by A5,Th57;
    then
A7: p,q // t,sum(q,s,o) by A1,A4,Th8;
    congr o,p,r,sum(p,r,o) by Def5;
    then congr p,o,sum(p,r,o),r by Th69;
    then congr p,q,sum(p,r,o),t by A2,Th70;
    then p,q // sum(p,r,o),t by Th57;
    then p,q // t,sum(p,r,o) by Th6;
    then t,sum(q,s,o) // t,sum(p,r,o) by A3,A7,Def1;
    then t,sum(q,s,o) // sum(p,r,o),sum(q,s,o) by Th7;
    hence thesis by A6,A7,Th8;
  end;
  hence thesis by Th3,Th85;
end;
