
theorem
   for D be set, f,g be XFinSequence of D, h be FinSequence of D holds
   (f^g)^h = f^(g^h) & (f^h)^g = f^(h^g) & (h^f)^g = h^(f^g)
   proof
     let D be set, f,g be XFinSequence of D, h be FinSequence of D;
L1:  (f^g)^h = (f^g)^(FS2XFS h) by XFF
     .= f^(g^(FS2XFS h)) by AFINSQ_1:27
     .= f^(g^h) by XFF;
     L2: (f^h)^g = f^(XFS2FS (FS2XFS h))^g
     .= f^(FS2XFS h)^g by XFX
     .= f^((FS2XFS h)^FS2XFS(XFS2FS g)) by AFINSQ_1:27
     .= f^FS2XFS (h^(XFS2FS g)) by SFF
     .= f^(h^(XFS2FS g)) by XFF
     .= f^(h^g) by FFX;
     (h^f)^g = (h^(FS2XFS(XFS2FS f)))^(XFS2FS(FS2XFS(XFS2FS g))) by FFX
     .= h^(XFS2FS f)^(XFS2FS g) by FFX
     .= h^((XFS2FS f)^(XFS2FS g)) by FINSEQ_1:32
     .= h^XFS2FS(f^g) by SXX
     .= h^(f^g) by FFX;
     hence thesis by L1,L2;
   end;
