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 Th37:
  Gen w,y implies for u,u9,u1,u2,v1,v2,t1,t2,w1,w2 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 & u,u9,v1,
w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2 // v1,v2 holds t1,t2 //
  w1,w2
proof
  assume
A1: Gen w,y;
  let u,u9,u1,u2,v1,v2,t1,t2,w1,w2 be VECTOR of V such that
A2: u<>u9 and
A3: u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y and
A4: u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y and
A5: u1,u2 // v1,v2;
  set A1 = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,u1))*(PProJ(w,y,u-u9,u-u9)
)", A2 = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,u2))*(PProJ(w,y,u-u9,u-u9))",
A3 = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,v1))*(PProJ(w,y,u-u9,u-u9))", A4
  = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,v2))*(PProJ(w,y,u-u9,u-u9))";
A6: (u1+A1*(u-u9))=t1 & (u2+A2*(u-u9))=t2 by A1,A2,A3,Th36;
A7: (v1+A3*(u-u9))=w1 & (v2+A4*(u-u9))=w2 by A1,A2,A4,Th36;
A8: t1=u1+A1*(u-u9) & t2=u2+A2*(u-u9) by A1,A2,A3,Th36;
  per cases;
  suppose
    u1=u2;
    then t1=t2 by A1,A2,A3,Th24;
    hence thesis by ANALOAF:def 1;
  end;
  suppose
    v1=v2;
    then w1=w2 by A1,A2,A4,Th24;
    hence thesis by ANALOAF:def 1;
  end;
  suppose that
A9: u1<>u2 & v1<>v2;
    set cc = (PProJ(w,y,u-u9,u-u9))";
    set vv = ((-2)*PProJ(w,y,u-u9,v2-v1));
    set uu = ((-2)*PProJ(w,y,u-u9,u2-u1));
A10: w2-w1 = (v2 - v1)+(A4-A3)*(u-u9) by A1,A7,Lm21;
    consider a,b such that
A11: a*(u2-u1) = b*(v2-v1) and
A12: 0<a & 0<b by A5,A9,ANALOAF:7;
A13: a*uu = (-2)*(a*PProJ(w,y,u-u9,u2-u1))
      .= (-2)*PProJ(w,y,u-u9,b*(v2-v1)) by A1,A11,Th31
      .= (-2)*(b*PProJ(w,y,u-u9,v2-v1)) by A1,Th31
      .=b*vv;
A14: a*(A2-A1) = a*(uu*cc) by A1,A8,Lm21
      .= (b*vv)*cc by A13,XCMPLX_1:4
      .= b*(vv*cc)
      .= b*(A4-A3) by A1,A7,Lm21;
    t2-t1 = (u2 - u1)+(A2-A1)*(u-u9) by A1,A6,Lm21;
    then a*(t2-t1) = a*(u2 - u1) + a*((A2-A1)*(u-u9)) by RLVECT_1:def 5
      .= b*(v2-v1) + (b*(A4-A3))*(u-u9) by A11,A14,RLVECT_1:def 7
      .= b*(v2 - v1) + b*((A4-A3)*(u-u9)) by RLVECT_1:def 7
      .= b*(w2-w1) by A10,RLVECT_1:def 5;
    hence thesis by A12,ANALOAF:def 1;
  end;
end;
