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;

theorem Th46:
  for POS being OrtAfSp for M,N being Subset of POS, M9,N9 being
  Subset of the AffinStruct of POS st M = M9 & N = N9
  holds M // N iff M9 // N9
proof
  let POS be OrtAfSp;
  let M,N be Subset of POS, M9,N9 be Subset of the AffinStruct of POS such that
A1: M = M9 & N = N9;
  hereby
    assume M // N;
    then consider a,b,c,d being Element of POS such that
A2: a<>b & c <>d and
A3: M = Line(a,b) & N = Line(c,d) and
A4: a,b // c,d;
    reconsider a9=a,b9=b,c9=c,d9=d as Element of the AffinStruct of POS;
A5: a9,b9 // c9,d9 by A4,Th36;
    M9=Line(a9,b9) & N9=Line(c9,d9) by A1,A3,Th41;
    hence M9 // N9 by A2,A5,AFF_1:37;
  end;
  assume M9 // N9;
  then consider a9,b9,c9,d9 being Element of the AffinStruct of POS such that
A6: a9<>b9 & c9<>d9 and
A7: a9,b9 // c9,d9 and
A8: M9 = Line(a9,b9) & N9 = Line(c9,d9) by AFF_1:37;
  reconsider a=a9,b=b9,c =c9,d=d9 as Element of POS;
A9: a,b // c,d by A7,Th36;
  M=Line(a,b) & N=Line(c,d) by A1,A8,Th41;
  hence thesis by A6,A9;
end;
