reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th36:
  for a,b being Real, A being non empty closed_interval Subset of REAL
   holds chi(A,A)=(Cst(1))|A
proof
  let a,b be Real, A be non empty closed_interval Subset of REAL;
  dom ((Cst(1))|A) = A & for x being object st x in A holds (x in A implies (
  (Cst(1))|A).x = 1) & (not x in A implies ((Cst(1))|A).x = 0)
  proof
A1: dom (Cst(1))=REAL & dom ((Cst(1))|A)=dom (Cst(1)) /\ A by FUNCOP_1:13
,RELAT_1:61;
    hence dom ((Cst(1))|A)=A by XBOOLE_1:28;
    let x be object;
    assume
A2: x in A;
    then x in dom ((Cst(1))|A) by A1,XBOOLE_0:def 4;
    then ((Cst 1)|A).x = (REAL-->1).x by FUNCT_1:47
      .= 1 by A2,FUNCOP_1:7;
    hence thesis by A2;
  end;
  hence thesis by FUNCT_3:def 3;
end;
