theorem Th7:
  for A being finite Subset of REAL holds SgmX(RealOrd,A) is increasing
proof
  let A be finite Subset of REAL;
  set IT = SgmX(RealOrd,A);
  let n,m be Nat such that
A1: n in dom IT & m in dom IT and
A2: n<m;
A3: RealOrd linearly_orders A by Th6,ORDERS_1:38;
  IT/.n = IT.n & IT/.m = IT.m by A1,PARTFUN1:def 6;
  then [IT.n,IT.m] in RealOrd by A1,A2,A3,PRE_POLY:def 2;
  then
A4: IT.n <= IT.m by Th4;
  IT is one-to-one by Th6,ORDERS_1:38,PRE_POLY:10;
  then IT.n <> IT.m by A1,A2,FUNCT_1:def 4;
  hence thesis by A4,XXREAL_0:1;
end;
