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
  p in M & p in N & a in M & b in N & a<>b & a in K & b in K & A _|_ M &
  A _|_ N & K is being_line implies A _|_ K
proof
  assume that
A1: p in M & p in N & a in M & b in N and
A2: a<>b and
A3: a in K & b in K and
A4: A _|_ M and
A5: A _|_ N and
A6: K is being_line;
  A is being_line by A4;
  then consider q,r such that
A7: q<>r and
A8: A = Line(q,r);
  reconsider q9=q,r9=r as Element of the AffinStruct of POS;
  Line(q,r) = Line(q9,r9) by Th41;
  then q in A & r in A by A8,AFF_1:15;
  then q,r _|_ p,a & q,r _|_ p,b by A1,A4,A5,Th56;
  then
A9: q,r _|_ a,b by Def7;
  K = Line(a,b) by A2,A3,A6,Th54;
  hence thesis by A2,A7,A8,A9,Th45;
end;
