reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem Th61:
  f= chi(X,C) iff ( dom f = C & for c holds (c in X implies f.c =
  1) & (not c in X implies f.c = 0))
proof
  thus f= chi(X,C) implies (dom f = C & for c holds (c in X implies f.c = 1) &
  (not c in X implies f.c = 0)) by FUNCT_3:def 3;
  assume that
A1: dom f = C and
A2: for c holds (c in X implies f.c = 1) & (not c in X implies f.c = 0);
  for x st x in C holds (x in X implies (f qua Function).x = 1) & (not x
  in X implies (f qua Function).x = 0) by A2;
  hence thesis by A1,FUNCT_3:def 3;
end;
