 reserve Omega for non empty set;
 reserve r for Real;
 reserve Sigma for SigmaField of Omega;
 reserve P for Probability of Sigma;

theorem Th1:
  for f be one-to-one Function, A,B be Subset of dom f st A misses B holds
  rng (f|A) misses rng (f|B)
  proof
    let f be one-to-one Function, A,B be Subset of dom f;
    assume A1:A misses B;
    assume rng (f|A) meets rng (f|B);then
    consider y be object such that
    A2: y in rng (f|A) & y in rng (f|B) by XBOOLE_0:3;
    consider xa be object such that
    A3: xa in dom (f|A) & y = (f|A).xa by A2,FUNCT_1:def 3;
    consider xb be object such that
    A4: xb in dom (f|B) & y = (f|B).xb by A2,FUNCT_1:def 3;
    A5:xa in dom f & xb in dom f by A3,A4,RELAT_1:57;
    y=f.xa &y=f.xb by A3,A4,FUNCT_1:47;then
    A6:xa=xb by A5,FUNCT_1:def 4;
    dom (f|A) c= A & dom (f|B) c=B by RELAT_1:58;
    then xa in A /\ B by A6,A3,A4,XBOOLE_0:def 4;
    hence contradiction by A1,XBOOLE_0:4;
  end;
