reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th30:
  1_F+1_F<>0.F & not p _|_ a & not p _|_ x & not q _|_ a & not q
_|_ x & not b _|_ a implies ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = ProJ(a,b,q)*
  ProJ(q,a,x)*ProJ(x,q,y)
proof
  set 0F = 0.F, 1F = 1_F;
  assume that
A1: 1_F+1_F<>0.F & not p _|_ a and
A2: not p _|_ x and
A3: not q _|_ a and
A4: not q _|_ x and
A5: not b _|_ a;
A6: now
A7: ProJ(a,b,q) <> 0F by A3,A5,Th23;
    assume
A8: not y _|_ x;
    ProJ(a,q,p)*ProJ(p,a,x) = ProJ(x,q,p)*ProJ(q,a,x) by A1,A2,A3,A4,Th29;
    then (ProJ(a,b,p)*ProJ(a,b,q)")*ProJ(p,a,x) = ProJ(x,q,p)*ProJ (q,a,x) by
A3,A5,Th24;
    then (ProJ(a,b,q)"*ProJ(a,b,p))*ProJ(p,a,x) = (ProJ(x,y,p)*ProJ(x,y,q)")*
    ProJ (q,a,x) by A4,A8,Th24;
    then
    ProJ(a,b,q)*(ProJ(a,b,q)"*(ProJ(a,b,p)*ProJ(p,a,x))) = ProJ(a,b,q)*((
    ProJ(x,y,p)*ProJ(x,y,q)")*ProJ (q,a,x)) by GROUP_1:def 3;
    then
    (ProJ(a,b,q)*ProJ(a,b,q)")*(ProJ(a,b,p)*ProJ(p,a,x)) = ProJ(a,b,q)*((
    ProJ(x,y,p)*ProJ(x,y,q)")*ProJ (q,a,x)) by GROUP_1:def 3;
    then
    (ProJ(a,b,p)*ProJ(p,a,x))*(1F) = ProJ(a,b,q)*((ProJ(x,y,p)*ProJ(x,y,q
    )")*ProJ (q,a,x)) by A7,VECTSP_1:def 10;
    then ProJ(a,b,p)*ProJ(p,a,x) = ProJ(a,b,q)*((ProJ(x,y,p)*ProJ(x,y,q)")*
    ProJ (q,a,x));
    then ProJ(a,b,p)*ProJ(p,a,x) = ProJ(a,b,q)*((ProJ(x,y,q)"*ProJ(x,p,y)")*
    ProJ (q,a,x)) by A2,A8,Th25;
    then ProJ(a,b,p)*ProJ(p,a,x) = (ProJ(a,b,q)*(ProJ(x,y,q)"*ProJ(x,p,y)"))*
    ProJ (q,a,x) by GROUP_1:def 3;
    then ProJ(a,b,p)*ProJ(p,a,x) = ProJ(q,a,x)*((ProJ(a,b,q)*ProJ(x,y,q)")*
    ProJ (x,p,y)") by GROUP_1:def 3;
    then
    ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = ((ProJ(q,a,x)*(ProJ(a,b,q)*ProJ
    (x,y,q)"))*ProJ(x,p,y)")*ProJ (x,p,y) by GROUP_1:def 3;
    then
A9: ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = (ProJ(q,a,x)*(ProJ(a,b,q)*ProJ(
    x,y,q)"))*(ProJ(x,p,y)"*ProJ (x,p,y)) by GROUP_1:def 3;
    ProJ(x,p,y) <> 0F by A2,A8,Th23;
    then
    ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = (ProJ(q,a,x)*(ProJ(a,b,q)*ProJ(
    x,y,q)"))*(1F) by A9,VECTSP_1:def 10;
    then
    ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = ProJ(q,a,x)*(ProJ(a,b,q)*ProJ(x
    ,y,q)");
    then
    ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = ProJ(q,a,x)*(ProJ(a,b,q)*ProJ(x
    ,q,y)) by A4,A8,Th25;
    hence thesis by GROUP_1:def 3;
  end;
  now
    assume
A10: y _|_ x;
    then ProJ(x,p,y) = 0F by A2,Th23;
    then
A11: ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = 0F;
    ProJ(x,q,y) = 0F by A4,A10,Th23;
    hence thesis by A11;
  end;
  hence thesis by A6;
end;
