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;

theorem Th28:
  for u,u1,u2,v1,v2,t1,t2,w1,w2 being VECTOR of V holds ( u=u1#t1
& u=u2#t2 & u=v1#w1 & u=v2#w2 & u1,u2,v1,v2 are_DTr_wrt w,y implies t1,t2,w1,w2
  are_DTr_wrt w,y)
proof
  let u,u1,u2,v1,v2,t1,t2,w1,w2 be VECTOR of V;
  assume that
A1: u=u1#t1 & u=u2#t2 and
A2: u=v1#w1 & u=v2#w2 and
A3: u1,u2,v1,v2 are_DTr_wrt w,y;
A4: u1,u2 // v1,v2 by A3;
  set p=u1#u2,q=v1#v2,r=t1#t2,s=w1#w2;
A5: q#s = u#u by A2,Th6
    .=u;
  p#r = u#u by A1,Th6
    .=u;
  then
A6: s-r = -(q-p) by A5,Lm3
    .= (-1)*(q-p) by RLVECT_1:16;
A7: u2-u1 = -(t2-t1) & v2-v1 = -(w2-w1) by A1,A2,Lm3;
A8: t1,t2 // w1,w2
  proof
    per cases;
    suppose
      u1=u2;
      then t1=t2 by A1,Th7;
      hence thesis by ANALOAF:9;
    end;
    suppose
      v1=v2;
      then w1=w2 by A2,Th7;
      hence thesis by ANALOAF:9;
    end;
    suppose
      u1<>u2 & v1<>v2;
      then consider a,b such that
A9:   0<a & 0<b and
A10:  a*(u2-u1)=b*(v2-v1) by A4,ANALOAF:def 1;
      a*(t2-t1) = a*(-(-(t2-t1))) by RLVECT_1:17
        .= -(b*(-(w2-w1))) by A7,A10,RLVECT_1:25
        .= b*(-(-(w2-w1))) by RLVECT_1:25
        .= b*(w2-w1) by RLVECT_1:17;
      hence thesis by A9,ANALOAF:def 1;
    end;
  end;
  w2-w1 = -(v2-v1) by A2,Lm3;
  then
A11: w2-w1 = (-1)*(v2-v1) by RLVECT_1:16;
  v1,v2,p,q are_Ort_wrt w,y by A3;
  then v2-v1,q-p are_Ort_wrt w,y by ANALMETR:def 3;
  then w2-w1,s-r are_Ort_wrt w,y by A6,A11,ANALMETR:6;
  then
A12: w1,w2,r,s are_Ort_wrt w,y by ANALMETR:def 3;
  t2-t1 = -(u2-u1) by A1,Lm3;
  then
A13: t2-t1 = (-1)*(u2-u1) by RLVECT_1:16;
  u1,u2,p,q are_Ort_wrt w,y by A3;
  then u2-u1,q-p are_Ort_wrt w,y by ANALMETR:def 3;
  then t2-t1,s-r are_Ort_wrt w,y by A6,A13,ANALMETR:6;
  then t1,t2,r,s are_Ort_wrt w,y by ANALMETR:def 3;
  hence thesis by A8,A12;
end;
