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 Th54:
  a in K & b in K & a<>b & K is being_line implies K =Line(a,b)
proof
  assume that
A1: a in K & b in K & a<>b and
A2: K is being_line;
  reconsider a9=a,b9=b as Element of the AffinStruct of POS;
  reconsider K9=K as Subset of the AffinStruct of POS;
  K9 is being_line by A2,Th43;
  then K9 = Line(a9,b9) by A1,AFF_1:57;
  hence thesis by Th41;
end;
