theorem Th10:
  for f1,i1,i2,j st 1<=i1 & i1<=i2 & i2<=len f1 holds
    mid(f1,i1,i2).(len mid(f1,i1,i2))=f1.i2
proof
  let f1,i1,i2,j;
  assume that
A1: 1<=i1 and
A2: i1<=i2 and
A3: i2<=len f1;
A4: i1<=len f1 by A2,A3,XXREAL_0:2;
A5: 1<=i2 by A1,A2,XXREAL_0:2;
  then len mid(f1,i1,i2)=i2-'i1+1 by A1,A2,A3,A4,Th117;
  then 1<=len mid(f1,i1,i2) by NAT_1:11;
  then
A6: mid(f1,i1,i2).(len mid(f1,i1,i2)) =f1.(len mid(f1,i1,i2)+i1-'1) by A1,A2,A3
,A5,A4,Th117
    .=f1.(i2-'i1+1+i1-'1) by A1,A2,A3,A5,A4,Th117;
  i2-'i1+1+i1=i2-i1+1+i1 by A2,XREAL_1:233
    .=i2+1;
  hence thesis by A6,NAT_D:34;
end;
