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 Th40:
  for POS being OrtAfSp for a,b,c being Element of POS, a9,b9,c9
being Element of the AffinStruct of POS st a=a9& b=b9 & c = c9
  holds (LIN a,b,c iff LIN a9,b9,c9)
proof
  let POS be OrtAfSp;
  let a,b,c be Element of POS, a9,b9,c9 be Element of the AffinStruct of POS
   such that
A1: a=a9 & b=b9 & c = c9;
  hereby
    assume LIN a,b,c;
    then a,b // a,c;
    then a9,b9 // a9,c9 by A1,Th36;
    hence LIN a9,b9,c9 by AFF_1:def 1;
  end;
  assume LIN a9,b9,c9;
  then a9,b9 // a9,c9 by AFF_1:def 1;
  then a,b // a,c by A1,Th36;
  hence thesis;
end;
