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;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem Th103:
  B is having_a_unity associative commutative having_an_inverseOp
implies
    (the_inverseOp_wrt B).(B.(d1,d2)) =
         B.((the_inverseOp_wrt B).d1,(the_inverseOp_wrt B).d2)
proof
  set I=the_inverseOp_wrt B;
  assume
A1: B is having_a_unity associative commutative having_an_inverseOp;
  B. (B.(d1,d2), B.(I.d1,I.d2)) = B. (B.(d2,d1), B.(I.d1,I.d2))
    by A1,BINOP_1:def 2
  .= B.(B.(B.(d2,d1),I.d1),I.d2) by A1,BINOP_1:def 3
  .= B.(B.(d2,B.(d1,I.d1)),I.d2) by A1,BINOP_1:def 3
  .= B.(B.(d2,the_unity_wrt B),I.d2) by A1,FINSEQOP:59
  .= B.(d2,I.d2) by A1,SETWISEO:15
  .= the_unity_wrt B by A1,FINSEQOP:59;
  hence thesis by A1,FINSEQOP:60;
end;
