reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;
reserve G for finite Graph,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Vertex of G;
reserve G for finite oriented Graph,
  P,Q for oriented Chain of G,
  W for Function of (the carrier' of G), Real>=0,
  v1,v2,v3,v4 for Vertex of G;
reserve f,g,h for Element of REAL*,
  r for Real;
reserve G for oriented Graph,
  v1,v2 for Vertex of G,
  W for Function of (the carrier' of G), Real>=0;

theorem Th41:
  dom f = dom ((repeat(Relax(n)*findmin(n))).i.f)
proof
  set R=Relax(n), M=findmin(n);
  defpred P[Nat] means dom f = dom ((repeat(R*M)).$1.f);
  dom ((repeat(R*M)).0 .f)= dom ((id (REAL*)).f) by Def2
    .= dom f;
  then
A1: P[0];
A2: for k st P[k] holds P[k+1] by Th37;
  for k holds P[k] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
