reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;
reserve
  v,v1,v2 for FinSequence of REAL,
  n,m,k for Nat,
  x for set;

theorem
  v is increasing implies for n,m st n in dom v & m in dom v & n<=m
  holds v.n <= v.m
proof
  assume
A1: v is increasing;
  let n,m such that
A2: n in dom v & m in dom v and
A3: n<=m;
  now
    per cases;
    suppose
      n=m;
      hence thesis;
    end;
    suppose
      n<>m;
      then n<m by A3,XXREAL_0:1;
      hence thesis by A1,A2,SEQM_3:def 1;
    end;
  end;
  hence thesis;
end;
