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
   M is commutative associative &
   A is commutative associative having_a_unity having_an_inverseOp &
   M is_distributive_wrt A
implies
  for E be Enumeration of bool {2}
    for CE be non-empty non empty FinSequence of D* st
        CE = SignGenOp(f,A,bool {2}) * E & len f = 2
     ex S be Element of Fin dom App CE st
        S is with_evenly_repeated_values-member &
       SignGenOp(f,M,A,{2}) = A $$ (S,M "**" App CE)
proof
  assume
A1: M is commutative associative &
    A is commutative associative having_a_unity having_an_inverseOp &
    M is_distributive_wrt A;
  let E be Enumeration of bool {2};
  let CE be non-empty non empty FinSequence of D* such that
A2: CE = SignGenOp(f,A,bool {2}) * E & len f = 2;
  consider S be Element of Fin dom App CE such that
A3: S = { <*1,1*>, <*2,2*>} and
A4: SignGenOp(f,M,A,{2}) = A $$ (S,M "**" App CE) by A2,A1,Th141;
  take S;
  thus S is with_evenly_repeated_values-member by A3;
  thus SignGenOp(f,M,A,{2}) = A $$ (S,M "**" App CE) by A4;
end;
