reserve F for Field;
reserve S for OrtSp of F;
reserve a,b,c,d,p,q,r,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th25:
  not p _|_ a & not p _|_ x & not q _|_ a & not q _|_ x implies
  ProJ(a,q,p)*ProJ(p,a,x) = ProJ(q,a,x)*ProJ(x,q,p)
proof
  set 0F = 0.F;
  set 1F = 1_F;
  assume that
A1: not p _|_ a and
A2: not p _|_ x and
A3: not q _|_ a and
A4: not q _|_ x;
A5: p <> 0.S & q <> 0.S by A1,A3,Th1;
A6: not a _|_ p by A1,Th2;
  a <> 0.S & x <> 0.S by A1,A2,Th1,Th2;
  then consider r such that
A7: not r _|_ p and
A8: not r _|_ q and
A9: not r _|_ a and
A10: not r _|_ x by A5,Def1;
A11: not q _|_ r by A8,Th2;
A12: not x _|_ r by A10,Th2;
A13: not p _|_ r by A7,Th2;
  then
A14: ProJ(r,q,p) <> 0F by A11,Th20;
A15: not a _|_ r by A9,Th2;
A16: not x _|_ q by A4,Th2;
A17: not x _|_ p by A2,Th2;
A18: not a _|_ q by A3,Th2;
  ProJ(a,q,p)*ProJ(p,a,x)*ProJ(q,a,x)" = (ProJ(a,r,p)*ProJ(a,r,q)")*ProJ(
  p,a,x)*ProJ(q,a,x)" by A3,A9,Th21
    .= (ProJ(a,r,p)*ProJ(a,r,q)")*(ProJ(p,r,x)*ProJ(p,r,a)")*ProJ (q,a,x)"
  by A6,A7,Th21
    .= (ProJ(a,r,p)*ProJ(a,r,q)")*(ProJ(p,r,x)*ProJ(p,r,a)")*ProJ (q,x,a) by
A18,A16,Th22
    .= (ProJ(a,r,p)*ProJ(a,r,q)")*(ProJ(p,r,x)*ProJ(p,r,a)")*(ProJ(q,r,a)*
  ProJ (q,r,x)") by A16,A8,Th21
    .= (ProJ(a,p,r)"*ProJ(a,r,q)")*(ProJ(p,r,x)*ProJ(p,r,a)")*(ProJ(q,r,a)*
  ProJ (q,r,x)") by A1,A9,Th22
    .= (ProJ(a,p,r)"*ProJ(a,r,q)")*(ProJ(p,r,x)*ProJ(p,a,r))*(ProJ(q,r,a)*
  ProJ (q,r,x)") by A6,A7,Th22
    .= (ProJ(a,p,r)"*ProJ(a,q,r))*(ProJ(p,r,x)*ProJ(p,a,r))*(ProJ(q,r,a)*
  ProJ (q,r,x)") by A3,A9,Th22
    .= (ProJ(a,p,r)"*ProJ(a,q,r))*(ProJ(p,r,x)*ProJ(p,a,r))*(ProJ(q,a,r)"*
  ProJ (q,r,x)") by A18,A8,Th22
    .= (ProJ(a,p,r)"*ProJ(a,q,r))*(ProJ(p,x,r)"*ProJ(p,a,r))*(ProJ(q,a,r)"*
  ProJ (q,r,x)") by A17,A7,Th22
    .= (ProJ(p,x,r)"*ProJ(p,a,r))*(ProJ(a,p,r)"*ProJ(a,q,r))*(ProJ(q,a,r)"*
  ProJ (q,x,r)) by A16,A8,Th22
    .= ProJ(p,x,r)"*((ProJ(a,p,r)"*ProJ(a,q,r))*ProJ(p,a,r))*(ProJ(q,a,r)"*
  ProJ (q,x,r)) by GROUP_1:def 3
    .= ProJ(p,x,r)"*((ProJ(a,p,r)"*ProJ(p,a,r))*ProJ(a,q,r))*(ProJ(q,a,r)"*
  ProJ (q,x,r)) by GROUP_1:def 3
    .= ProJ(p,x,r)"*(ProJ(r,a,p)*ProJ(a,q,r))*(ProJ(q,a,r)"*ProJ (q,x,r)) by A1
,A9,Th24
    .= (ProJ(p,x,r)"*ProJ(r,a,p))*ProJ(a,q,r)*(ProJ(q,a,r)"*ProJ (q,x,r)) by
GROUP_1:def 3
    .= (ProJ(p,x,r)"*ProJ(r,a,p))*(ProJ(a,q,r)*(ProJ(q,a,r)"*ProJ (q,x,r)))
  by GROUP_1:def 3
    .= (ProJ(p,x,r)"*ProJ(r,a,p))*((ProJ(q,a,r)"*ProJ(a,q,r))*ProJ(q,x,r))
  by GROUP_1:def 3
    .= (ProJ(p,x,r)"*ProJ(r,a,p))*(ProJ(r,q,a)*ProJ(q,x,r)) by A18,A8,Th24
    .= ProJ(p,x,r)"*(ProJ(r,a,p)*(ProJ(r,q,a)*ProJ(q,x,r))) by GROUP_1:def 3
    .= ProJ(p,x,r)"*((ProJ(r,a,p)*ProJ(r,q,a))*ProJ(q,x,r)) by GROUP_1:def 3
    .= ProJ(p,x,r)"*((ProJ(r,a,p)*ProJ(r,a,q)")*ProJ(q,x,r)) by A11,A15,Th22
    .= ProJ(p,x,r)"*(ProJ(r,q,p)*ProJ(q,x,r)) by A11,A15,Th21
    .= ProJ(p,r,x)*(ProJ(r,q,p)*ProJ(q,x,r)) by A17,A7,Th22
    .= (ProJ(r,x,p)"*ProJ(x,r,p))*(ProJ(r,q,p)*ProJ(q,x,r)) by A13,A12,Th24
    .= (ProJ(r,x,p)"*ProJ(x,r,p))*(ProJ(r,q,p)*(ProJ(x,r,q)"*ProJ (r,x,q)))
  by A4,A10,Th24
    .= (ProJ(r,x,p)"*ProJ(x,r,p))*(ProJ(r,x,q)*(ProJ(r,q,p)*ProJ(x,r,q)"))
  by GROUP_1:def 3
    .= ((ProJ(x,r,p)*ProJ(r,x,p)")*ProJ(r,x,q))*(ProJ(r,q,p)*ProJ(x,r,q)")
  by GROUP_1:def 3
    .= (ProJ(x,r,p)*(ProJ(r,x,q)*ProJ(r,x,p)"))*(ProJ(r,q,p)*ProJ(x,r,q)")
  by GROUP_1:def 3
    .= (ProJ(x,r,p)*ProJ(r,p,q))*(ProJ(r,q,p)*ProJ(x,r,q)") by A13,A12,Th21
    .= ProJ(x,r,p)*(ProJ(r,p,q)*(ProJ(r,q,p)*ProJ(x,r,q)")) by GROUP_1:def 3
    .= ProJ(x,r,p)*((ProJ(r,p,q)*ProJ(r,q,p))*ProJ(x,r,q)") by GROUP_1:def 3
    .= ProJ(x,r,p)*((ProJ(r,q,p)"*ProJ(r,q,p))*ProJ(x,r,q)") by A13,A11,Th22
    .= ProJ(x,r,p)*(ProJ(x,r,q)"*(1F)) by A14,VECTSP_1:def 10
    .= ProJ(x,r,p)*ProJ(x,r,q)"
    .= ProJ(x,q,p) by A4,A10,Th21;
  then
A19: (ProJ(q,a,x)*ProJ(q,a,x)")*(ProJ(a,q,p)*ProJ(p,a,x)) = ProJ(q,a,x)*
  ProJ (x,q,p) by GROUP_1:def 3;
  ProJ(q,a,x) <> 0F by A18,A16,Th20;
  then (ProJ(a,q,p)*ProJ(p,a,x))*(1F) = ProJ(q,a,x)*ProJ (x,q,p) by A19,
VECTSP_1:def 10;
  hence thesis;
end;
