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
  a in M & b in M & a<>b & M is being_line & c in N & d in N & c <>d & N
  is being_line & a,b // c,d implies M // N
proof
  assume that
A1: a in M & b in M and
A2: a<>b and
A3: M is being_line & c in N & d in N and
A4: c <>d and
A5: N is being_line and
A6: a,b // c,d;
  M = Line(a,b) & N = Line(c,d) by A1,A2,A3,A4,A5,Th54;
  hence thesis by A2,A4,A6;
end;
