reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem Th36:
  F is_distributive_wrt G implies
    F[:](f,G.(d1,d2)) = G.:(F[:](f,d1),F[:] (f,d2))
proof
  assume
A1: F is_distributive_wrt G;
  now
    let c;
    thus (F[:](f,G.(d1,d2))).c = F.(f.c,G.(d1,d2)) by FUNCOP_1:48
      .= G.(F.(f.c,d1),F.(f.c,d2)) by A1,BINOP_1:11
      .= G.(F[:](f,d1).c,F.(f.c,d2)) by FUNCOP_1:48
      .= G.((F[:](f,d1)).c,(F[:](f,d2)).c) by FUNCOP_1:48
      .= (G.:(F[:](f,d1),F[:](f,d2))).c by FUNCOP_1:37;
  end;
  hence thesis by FUNCT_2:63;
end;
