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 Th39:
  Gen w,y implies for u,u9,u1,u2,t1,t2 being VECTOR of V st u<>u9
& u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y holds u,u9,u1#u2,t1#
  t2 are_DTr_wrt w,y
proof
  assume
A1: Gen w,y;
  let u,u9,u1,u2,t1,t2 be VECTOR of V such that
A2: u<>u9 and
A3: u,u9,u1,t1 are_DTr_wrt w,y and
A4: u,u9,u2,t2 are_DTr_wrt w,y;
  u1,t1,u2,t2 are_DTr_wrt w,y by A1,A2,A3,A4,Th19;
  then
A5: u1,t1,u1#u2,t1#t2 are_DTr_wrt w,y by A1,Th26;
  u2,t2,u1,t1 are_DTr_wrt w,y by A1,A2,A3,A4,Th19;
  then
A6: u2,t2,u2#u1,t2#t1 are_DTr_wrt w,y by A1,Th26;
  per cases;
  suppose
A7: u1<>t1;
    u1,t1,u,u9 are_DTr_wrt w,y by A3,Th21;
    hence thesis by A1,A5,A7,Th19;
  end;
  suppose
A8: u2<>t2;
    u2,t2,u,u9 are_DTr_wrt w,y by A4,Th21;
    hence thesis by A1,A6,A8,Th19;
  end;
  suppose that
A9: u1=t1 and
A10: u2=t2;
    u#u9,(u1#u2)#(t1#t2),u1#u2,t1#t2 are_Ort_wrt w,y by A1,A9,A10,Lm6;
    then
A11: u1#u2,t1#t2,u#u9,(u1#u2)#(t1#t2) are_Ort_wrt w,y by Lm5;
A12: u,u9,u#u9,u1#u2 are_Ort_wrt w,y
    proof
      set uu9=u#u9;
A13:  2*(u1#u2) = (1+1)*(u1#u2) .= 1*(u1#u2)+1*(u1#u2) by RLVECT_1:def 6
        .= (u1#u2)+1*(u1#u2) by RLVECT_1:def 8
        .= (u1#u2)+(u1#u2) by RLVECT_1:def 8
        .=u1+u2 by Def2;
A14:  -(2*uu9) = -((1+1)*uu9) .= -(1*uu9+1*uu9) by RLVECT_1:def 6
        .= -(uu9 + 1*uu9) by RLVECT_1:def 8
        .= -(uu9+uu9) by RLVECT_1:def 8
        .= -uu9 + -uu9 by Lm19;
      u,u9,uu9,u2#t2 are_Ort_wrt w,y by A4;
      then u9-u,u2-uu9 are_Ort_wrt w,y by A10,ANALMETR:def 3;
      then
A15:  u9-u,(2")*(u2-uu9) are_Ort_wrt w,y by ANALMETR:7;
      u,u9,uu9,u1#t1 are_Ort_wrt w,y by A3;
      then u9-u,u1-uu9 are_Ort_wrt w,y by A9,ANALMETR:def 3;
      then
A16:  u9-u,(2")*(u1-uu9) are_Ort_wrt w,y by ANALMETR:7;
      u1#u2-uu9 = (2"*2)*(u1#u2-uu9) by RLVECT_1:def 8
        .= 2"*(2*(u1#u2-uu9)) by RLVECT_1:def 7
        .= 2"*((u1+u2)-(2*uu9)) by A13,RLVECT_1:34
        .= 2"*(((u1+u2)+ -uu9) + -uu9) by A14,RLVECT_1:def 3
        .= 2"*(((u1-uu9)+u2) + -uu9) by RLVECT_1:def 3
        .= 2"*((u1-uu9)+(u2-uu9)) by RLVECT_1:def 3
        .= 2"*(u1-uu9)+2"*(u2-uu9) by RLVECT_1:def 5;
      then u9-u,u1#u2-uu9 are_Ort_wrt w,y by A1,A16,A15,ANALMETR:10;
      hence thesis by ANALMETR:def 3;
    end;
    u,u9 // u1#u2,t1#t2 by A9,A10,ANALOAF:9;
    hence thesis by A9,A10,A11,A12;
  end;
end;
