gG:=(x+y)/2-(x-y)/2 F+F^2/2-1
gOne:=(x+y)/2-1
gF:=(x-y)/2-F
fG:=(x+y)/2F-(x-y)/2
fOne:=-(x-y)/2
fF:=1-(x+y)/2
gH:=-((-x-y+2 +2 H  )/(2  ))
fH:=-((x-y-2 F  )/(2 ))
y:=x^(-1)
fPhi:=-(FH(x+y)-HOne(x-y))/2
gPhi:=-(FH(y-x)+HOne(x+y)+2Phi)/2
fPsi:=-(FF(x+y)-FOne(x-y))/2
gPsi:=-(FF(y-x)+FOne(x+y)+2Psi)/2
psiPhi:=-gH-(g+gg)(Phi+gPhi)+fg fPhi
psiPsi:=-gF-(Psi+gPsi)(g+gg)+fPsi fg
psiOne:=-(g+gg)(1+gOne)+fg fOne
psiG:=-gg-(G+gG)(g+gg)+fg fG
psiF:=-fg-(F+gF)(g+gg)+fg fF
intphi1:=-t g+G-F

(*Product*)

intphi1:=-t g+G-F
psi1phi2:=t psiG-t F psiOne+psiPhi+Psi psiOne+psiPsi
psi2phi3:=fOne+fF

psi1phi3:=t fG-t F fOne+fPhi+Psi fOne+fPsi
psi2phi2:=psiOne+psiF

Collect[Simplify[(1-intphi1-psi1phi2)(1-psi2phi3)-psi1phi3 psi2phi2],x]

G:=-H

(*Using Now 
G=-H
F^2/2=-G,
we then obtain
*)

Simplify[-FF g-FH g-FOne g-FF gg-FH gg-FOne gg-g HOne-gg HOne+g Psi+gg Psi+g t+gg t]

(*Note that here 
g+gg:=(SuperMinus[g])+<g|g> 
and also
FF+FH+FOne+HOne=<(F+1)|(F+H)>,
we then obtain the result.
*)