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 Th48:
  g=(repeat(Relax(n)*findmin(n))).1.f & h=(repeat(Relax(n)*findmin
  (n))).i.f & 1<=i & i <= LifeSpan(Relax(n)*findmin(n),f,n) & m in UsedVx(g,n)
  implies m in UsedVx(h,n)
proof
  set RF=Relax(n)*findmin(n), RT=repeat RF, cn=LifeSpan(RF,f,n);
  assume that
A1: g=RT.1.f and
A2: h=RT.i.f & 1<=i & i <= cn and
A3: m in UsedVx(g,n);
  defpred P[Nat] means
   1<=$1 & $1<=cn implies m in UsedVx(RT.$1.f,n);
A4: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A5: P[k];
    hereby
      assume that
      1<=k+1 and
A6:   k+1 <= cn;
      per cases;
      suppose
        k=0;
        hence m in UsedVx(RT.(k+1).f,n) by A1,A3;
      end;
      suppose
A7:     k<>0;
        k < cn by A6,NAT_1:13;
        then OuterVx(RT.k.f,n) <> {} by Def4;
        then
        UsedVx(RT.(k+1).f,n)=UsedVx(RT.k.f,n) \/ {Argmin(OuterVx(RT.k.f,n
        ),RT.k.f,n)} by Th39;
        then
A8:     UsedVx(RT.k.f,n) c= UsedVx(RT.(k+1).f,n) by XBOOLE_1:7;
        k >= 1+ 0 by A7,INT_1:7;
        hence m in UsedVx(RT.(k+1).f,n) by A5,A6,A8,NAT_1:13;
      end;
    end;
  end;
A9: P[0];
  for k holds P[k] from NAT_1:sch 2(A9,A4);
  hence thesis by A2;
end;
