reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;
reserve AS for non empty AffinStruct;
reserve S for OAffinPlane;
reserve x,y,z,t,u for Element of S;

theorem Th43:
  not x,y '||' z,t implies ex u st x,y '||' x,u & z,t '||' z,u
proof
  assume not x,y '||' z,t;
  then ( not x,y // z,t)& not x,y // t,z;
  then consider u such that
A1: ( x,y // x,u or x,y // u,x)&( z,t // z,u or z,t // u,z) by ANALOAF:def 6;
  x,y '||' x,u & z,t '||' z,u by A1;
  hence thesis;
end;
