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 Th41:
  for POS being OrtAfSp for a,b being Element of POS, a9,b9 being
  Element of the AffinStruct of POS st a=a9 & b=b9
   holds Line(a,b) = Line(a9,b9)
proof
  let POS be OrtAfSp;
  let a,b be Element of POS, a9,b9 be Element of the AffinStruct of POS
   such that
A1: a=a9 & b=b9;
  set X = Line(a,b), Y = Line(a9,b9);
  now
    let x1 be object;
A2: now
      assume
A3:   x1 in Y;
      then reconsider x9=x1 as Element of the AffinStruct of POS;
      reconsider x=x9 as Element of POS;
      LIN a9,b9,x9 by A3,AFF_1:def 2;
      then LIN a,b,x by A1,Th40;
      hence x1 in X by Def10;
    end;
    now
      assume
A4:   x1 in X;
      then reconsider x=x1 as Element of POS;
      reconsider x9=x as Element of the AffinStruct of POS;
      LIN a,b,x by A4,Def10;
      then LIN a9,b9,x9 by A1,Th40;
      hence x1 in Y by AFF_1:def 2;
    end;
    hence x1 in X iff x1 in Y by A2;
  end;
  hence thesis by TARSKI:2;
end;
