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 Th46:
  f, (repeat(Relax(n)*findmin(n))).i.f equal_at 3*n+1,n*n+3*n
proof
  set R=Relax(n), M=findmin(n), m=3*n+1, mm=n*n+3*n;
  defpred P[Nat] means f,(repeat(R*M)).$1.f equal_at m,mm;
A1: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A2: P[k];
    (repeat(R*M)).k.f, (repeat(R*M)).(k+1).f equal_at m,mm by Th44;
    hence thesis by A2,Th43;
  end;
  (repeat(R*M)).0 .f =f by Th21;
  then
A3: P[0] by Th42;
  for k holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
