reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem Th17:
  x1 in A implies
  (RealFuncZero A) +* (x1 .--> 1) in Funcs (A,REAL) &
  (RealFuncUnit A) +* (x1 .--> 0) in Funcs (A,REAL)
  proof
    assume a0: x1 in A;
A2: dom ((RealFuncZero A) +* (x1 .--> 1)) =
       dom RealFuncZero A \/ dom (x1 .--> 1) by FUNCT_4:def 1
       .= dom RealFuncZero A \/ {x1}
       .= A \/ {x1}
       .= A by a0,XBOOLE_1:12,ZFMISC_1:31;
a2: dom ((RealFuncUnit A) +* (x1 .--> 0)) =
       dom RealFuncUnit A \/ dom (x1 .--> 0) by FUNCT_4:def 1
       .= dom RealFuncUnit A \/ {x1}
       .= A \/ {x1}
       .= A by a0,XBOOLE_1:12,ZFMISC_1:31;
H2: rng ((RealFuncZero A) +* (x1 .--> 1)) c= rng RealFuncZero A \/
      rng (x1 .--> 1) by FUNCT_4:17;
B1: rng RealFuncZero A c= REAL by RELAT_1:def 19;
    rng (x1 .--> 1) = {1} by FUNCOP_1:8; then
    rng (x1 .--> 1) c= REAL by XREAL_0:def 1,ZFMISC_1:31; then
    rng RealFuncZero A \/ rng (x1 .--> 1) c= REAL \/ REAL by B1,XBOOLE_1:13;
      then
S1: (RealFuncZero A) +* (x1 .--> 1) is Function of A, REAL by A2,FUNCT_2:2,
      H2,XBOOLE_1:1;
H2: rng ((RealFuncUnit A) +* (x1 .--> 0)) c= rng RealFuncUnit A \/
      rng (x1 .--> 0) by FUNCT_4:17;
B1: rng RealFuncUnit A c= REAL by RELAT_1:def 19;
    rng (x1 .--> 0) = {0} by FUNCOP_1:8; then
    rng (x1 .--> 0) c= REAL by XREAL_0:def 1,ZFMISC_1:31; then
    rng RealFuncUnit A \/ rng (x1 .--> 0) c= REAL \/ REAL by B1,XBOOLE_1:13;
      then
    (RealFuncUnit A) +* (x1 .--> 0) is Function of A, REAL
      by a2,FUNCT_2:2,H2,XBOOLE_1:1;
    hence thesis by S1,FUNCT_2:8;
  end;
