reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);
reserve x,a,b,c,d,p,q,y for Element of POS;
reserve A,K,M for Subset of POS;
reserve POS for OrtAfSp;
reserve A,K,M,N for Subset of POS;
reserve a,b,c,d,p,q,r,s for Element of POS;
reserve POS for OrtAfPl;
reserve K,M,N for Subset of POS;
reserve x,a,b,c,d,p,q for Element of POS;

theorem
  K is being_line implies ex x st a,x _|_ K & x in K
proof
  assume K is being_line;
  then consider p,q such that
A1: p<>q and
A2: K = Line(p,q);
  consider x such that
A3: a,x _|_ p,q and
A4: LIN p,q,x by Th69;
  take x;
  thus a,x _|_ K by A1,A2,A3;
  thus thesis by A2,A4,Def10;
end;
