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 Th26:
  F is associative & (i>=1 & j>=1 or F is having_a_unity) implies
  F"**"((i+j)|->d) = F.(F"**"(i|->d),F"**"(j|->d))
proof
  assume
A1: F is associative;
  set p1 = (i|->d),p2 = (j|->d);
  assume i>=1 & j>=1 or F is having_a_unity;
  then len p1 >= 1 & len p2 >= 1 or F is having_a_unity by CARD_1:def 7;
  then F "**"(p1^p2) = F.(F"**"p1,F"**"p2) by A1,FINSOP_1:5;
  hence thesis by FINSEQ_2:123;
end;
