reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

theorem Th1:
  F is commutative associative & c1 <> c2 implies F $$ ({.c1,c2.},f
  ) = F.(f.c1, f.c2)
proof
  assume that
A1: F is commutative associative and
A2: c1 <> c2;
  consider g being Function of Fin C, D such that
A3: F $$ ({.c1,c2.},f) = g.{c1,c2} and
  for e st e is_a_unity_wrt F holds g.{} = e and
A4: for c holds g.{c} = f.c and
A5: for B st B c= { c1,c2 } & B <> {} for c st c in { c1,c2 } \ B holds
  g.(B \/ {c}) = F.(g.B,f.c) by A1,SETWISEO:def 3;
  c1 in {c1} & not c2 in {c1} by A2,TARSKI:def 1;
  then {c1,c2} \ {c1} = {c2} by ZFMISC_1:62;
  then
A6: c2 in {c1,c2} \ {c1} by TARSKI:def 1;
  thus F $$ ({.c1,c2.},f) = g.({.c1.} \/ {.c2.}) by A3,ENUMSET1:1
    .= F.(g.{c1}, f.c2) by A5,A6,ZFMISC_1:7
    .= F.(f.c1, f.c2) by A4;
end;
