reserve a,a1,a2,v,v1,v2,x for object;
reserve V,A for set;
reserve m,n for Nat;
reserve S,S1,S2 for FinSequence;

theorem Th25:
  S IsNDRankSeq V,A & m <= n & n in dom S implies S.m c= S.n
  proof
    assume that
A1: S IsNDRankSeq V,A and
A2: m <= n and
A3: n in dom S;
    per cases;
    suppose
A4:   m <> 0;
      then
A5:   0+1 <= m by NAT_1:13;
      n <= len S by A3,FINSEQ_3:25;
      then m <= len S by A2,XXREAL_0:2;
      then S.m = FNDSC(V,A).m & S.n = FNDSC(V,A).n
      by A1,A3,A5,Th19,FINSEQ_3:25;
      hence thesis by A2,A4,Th13;
    end;
    suppose
A6:   m = 0;
      not 0 in dom S by FINSEQ_3:24;
      hence thesis by A6,FUNCT_1:def 2;
    end;
  end;
