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;

theorem
  a in K & b in K & a<>b & K is being_line & (a,b _|_ c,d or c,d _|_ a,b
  ) implies c,d _|_ K
proof
  assume that
A1: a in K & b in K and
A2: a<>b and
A3: K is being_line &( a,b _|_ c,d or c,d _|_ a,b);
  c,d _|_ a,b & K = Line(a,b) by A1,A2,A3,Def7,Th54;
  hence thesis by A2;
end;
