reserve x,y for object,
        i,j,k,m,n for Nat;

theorem Th1:
  for f,g be non-decreasing ext-real-valued FinSequence st
    f.len f <= g.1 holds f^g is non-decreasing
proof
  let f,g be non-decreasing ext-real-valued FinSequence such that
  A1:f.len f <= g.1;
  set fg=f^g;
  for e1,e2 be ExtReal st e1 in dom fg & e2 in dom fg & e1 <= e2 holds
    fg.e1 <= fg.e2
  proof
    let e1,e2 be ExtReal such that A2:e1 in dom fg & e2 in dom fg & e1 <= e2;
    per cases;
    suppose A3:e1 in dom f & e2 in dom f;
      then fg.e1=f.e1 & fg.e2=f.e2 by FINSEQ_1:def 7;
      hence thesis by A2,A3,VALUED_0:def 15;
    end;
    suppose A4:e1 in dom f & not e2 in dom f;
      then consider i such that
      A5:i in dom g & e2=len f+ i by A2,FINSEQ_1:25;
      A6:1 <= e1 & e1 <= len f & 1 <= i & i <= len g by A4,A5,FINSEQ_3:25;
      then 1 <= len f & 1 <= len g by XXREAL_0:2;
      then len f in dom f & 1 in dom g by FINSEQ_3:25;
      then A7:f.e1 <= f.len f & g.1 <= g.i by VALUED_0:def 15,A4,A5,A6;
      then A8: f.e1 <= g.1 by A1,XXREAL_0:2;
      fg.e1 = f.e1 & fg.e2 = g.i by A5,A4,FINSEQ_1:def 7;
      hence thesis by A8,A7,XXREAL_0:2;
    end;
    suppose not e1 in dom f & e2 in dom f;
      then not (1<= e1 & e1 <= len f) & e2 <= len f & 1<= e1 by A2,FINSEQ_3:25;
      hence thesis by XXREAL_0:2,A2;
    end;
    suppose A9:not e1 in dom f & not e2 in dom f;
      then consider i such that
      A10:i in dom g & e1=len f+ i by A2,FINSEQ_1:25;
      consider j such that
      A11:j in dom g & e2=len f+ j by A2,FINSEQ_1:25,A9;
      fg.e1= g.i & fg.e2 = g.j by A10,A11,FINSEQ_1:def 7;
      hence thesis by A10,A2,A11,XREAL_1:6,VALUED_0:def 15;
    end;
  end;
  hence thesis by VALUED_0:def 15;
end;
