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 Th43:
  for POS being OrtAfSp for X being Subset of POS, Y being Subset
  of the AffinStruct of POS st X=Y holds X is being_line iff Y is being_line
proof
  let POS be OrtAfSp;
  let X be Subset of the carrier of POS, Y be Subset of the AffinStruct of POS
   such that
A1: X=Y;
  hereby
    assume X is being_line;
    then consider a,b being Element of POS such that
A2: a<>b and
A3: X = Line(a,b);
    reconsider a9=a,b9=b as Element of the AffinStruct of POS;
    Y = Line(a9,b9) by A1,A3,Th41;
    hence Y is being_line by A2,AFF_1:def 3;
  end;
  assume Y is being_line;
  then consider a9,b9 being Element of the AffinStruct of POS such that
A4: a9<>b9 and
A5: Y = Line(a9,b9) by AFF_1:def 3;
  reconsider a=a9,b=b9 as Element of POS;
  X = Line(a,b) by A1,A5,Th41;
  hence thesis by A4;
end;
