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 Th35:
  F is_distributive_wrt G implies
    F[;](G.(d1,d2),f) = G.:(F[;](d1,f),F[;] (d2,f))
proof
  assume
A1: F is_distributive_wrt G;
  now
    let c;
    thus (F[;](G.(d1,d2),f)).c = F.(G.(d1,d2),f.c) by FUNCOP_1:53
      .= G.(F.(d1,f.c),F.(d2,f.c)) by A1,BINOP_1:11
      .= G.(F[;](d1,f).c,F.(d2,f.c)) by FUNCOP_1:53
      .= G.((F[;](d1,f)).c,(F[;](d2,f)).c) by FUNCOP_1:53
      .= (G.:(F[;](d1,f),F[;](d2,f))).c by FUNCOP_1:37;
  end;
  hence thesis by FUNCT_2:63;
end;
