theorem Th87:
  for E be Enumeration of F st
    B is having_a_unity associative commutative having_an_inverseOp &
    1+len f in meet F & 2+len f in meet F
  holds
    B "**"(SignGenOp(f^<*d1*>^<*d2*>,B,F)*E) =
      B"**"(SignGenOp(f^<*B.(d1,d2)*>,B,F)*E)
proof
  set I=the_inverseOp_wrt B;
  let E be Enumeration of F such that
A1:  B is having_a_unity associative commutative having_an_inverseOp and
A2:  1+len f in meet F & 2+len f in meet F;
A3:  len (f^<*d1*>) = 1+len f by FINSEQ_2:16;
  then 1+len (f^<*d1*>) in meet F by A2;
  hence B "**" (SignGenOp(f^<*d1*>^<*d2*>,B,F)*E)
    = B[:](B "**" SignGenOp(f^<*d1*>,B,F)*E,I.d2) by NAT_1:11,Th83,A3
   .= B[:](B[:](B "**" SignGenOp(f,B,F)*E,I.d1),I.d2) by A1,Th83,A2
   .= B[:](B "**" SignGenOp(f,B,F)*E,B.(I.d1,I.d2)) by A1,FUNCOP_1:63
   .= B[:](B "**" SignGenOp(f,B,F)*E,I.(B.(d1,I.(I.d2)))) by A1,Th2
   .= B[:](B "**" SignGenOp(f,B,F)*E,I.(B.(d1,d2))) by A1,FINSEQOP:62
   .= B"**" (SignGenOp(f^<*B.(d1,d2)*>,B,F)*E) by A1,Th83,A2;
end;
