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 Th118:
  A is having_a_unity
implies
  for E be Enumeration of F
    for s be FinSequence st F={} & s in doms (SignGenOp(f,B,F) * E) holds
     (A "**" App (SignGenOp(f,B,F) * E)).s = the_unity_wrt A
proof
  assume
A1: A is having_a_unity;
  let E be Enumeration of F;
  let s be FinSequence such that
A2: F={} & s in doms (SignGenOp(f,B,F) * E);
  len s = len (SignGenOp(f,B,F) * E) = len E = card F by A2,Th47;
  then
A3: SignGenOp(f,B,F) * E={} & s={} by A2;
  s in dom App (SignGenOp(f,B,F) * E) by A2,Def9;
  hence (A "**" App (SignGenOp(f,B,F) * E)).s = A"**" <*>D by A3,Th59,Def10
  .= the_unity_wrt A by FINSOP_1:10,A1;
end;
