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
  F is commutative associative & c1 <> c2 & c1 <> c3 & c2 <> c3 implies
  F $$ ({.c1,c2,c3.},f) = F.(F.(f.c1, f.c2),f.c3)
proof
  assume that
A1: F is commutative associative and
A2: c1 <> c2;
  assume c1 <> c3 & c2 <> c3;
  then
A3: not c3 in {c1,c2} by TARSKI:def 2;
  thus F $$ ({.c1,c2,c3.},f) = F $$ ({.c1,c2.} \/ {.c3.},f) by ENUMSET1:3
    .= F.(F $$ ({.c1,c2.},f),f.c3) by A1,A3,Th2
    .= F.(F.(f.c1, f.c2),f.c3) by A1,A2,Th1;
end;
