reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th2:
for D be non empty set, d1,d2 be Element of D, B be BinOp of D st
  B is having_a_unity associative commutative having_an_inverseOp holds
    B.((the_inverseOp_wrt B).d1,d2) =
       (the_inverseOp_wrt B).(B.(d1,(the_inverseOp_wrt B).d2))
  &
    B.(d1,(the_inverseOp_wrt B).d2) =
       (the_inverseOp_wrt B).(B.((the_inverseOp_wrt B).d1,d2))
proof
  let D be non empty set, d1,d2 be Element of D, B be BinOp of D;
  set I=the_inverseOp_wrt B;
  assume
A1: B is having_a_unity associative commutative having_an_inverseOp; then
  B.(B.(d1,I.d2), B.(I.d1,d2))
     = B.(B.(I.d2,d1), B.(I.d1,d2)) by BINOP_1:def 2
    .= B.(B.(B.(I.d2,d1),I.d1),d2) by A1,BINOP_1:def 3
    .= B.(B.(I.d2,B.(d1,I.d1)),d2) by A1,BINOP_1:def 3
    .= B.(B.(I.d2,the_unity_wrt B),d2) by A1,FINSEQOP:59
    .= B.(I.d2,d2) by A1,SETWISEO:15
    .= the_unity_wrt B by A1,FINSEQOP:59;
  hence thesis by A1,FINSEQOP:60;
end;
