reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;
reserve p,p1,p2 for Point of TOP-REAL N;
reserve M for non empty MetrSpace;
reserve X,Y for non empty TopSpace;

theorem Th45:
  for f being Function of X,Y, P being non empty Subset of Y st X
is compact & Y is T_2 & f is continuous one-to-one & P=rng f holds ex f1 being
  Function of X,(Y|P) st f=f1 & f1 is being_homeomorphism
proof
  let f be Function of X,Y,P be non empty Subset of Y such that
A1: X is compact and
A2: Y is T_2 and
A3: f is continuous one-to-one and
A4: P=rng f;
  the carrier of (Y|P)=P & dom f=the carrier of X by FUNCT_2:def 1,PRE_TOPC:8;
  then reconsider f2=f as Function of X,(Y|P) by A4,FUNCT_2:1;
A5: dom f2=[#]X & f2 is continuous by A3,Th44,FUNCT_2:def 1;
  rng f2=[#](Y|P) & Y|P is T_2 by A2,A4,PRE_TOPC:def 5,TOPMETR:2;
  hence thesis by A1,A3,A5,COMPTS_1:17;
end;
