
theorem Th62:
  for A be Interval holds pre-Meas*<*A*> = <*pre-Meas.A*>
proof
    let A be Interval;
A1: A in Family_of_Intervals by MEASUR10:def 1;
 rng <*A*> = {A} by FINSEQ_1:38; then
    reconsider FA = <*A*> as FinSequence of Family_of_Intervals
     by A1,ZFMISC_1:31,FINSEQ_1:def 4;
    dom pre-Meas = Family_of_Intervals
  & rng FA c= Family_of_Intervals by FUNCT_2:def 1; then
    dom(pre-Meas*FA) = dom FA by RELAT_1:27; then
A2: dom(pre-Meas*FA) = Seg 1 by FINSEQ_1:38; then
A3: dom(pre-Meas*FA) = dom <*pre-Meas.A*> by FINSEQ_1:38;
    for n be Nat st n in dom(pre-Meas*FA) holds
      (pre-Meas*FA).n = <*pre-Meas.A*>.n
    proof
     let n be Nat;
     assume A4: n in dom(pre-Meas*FA); then
A5:  n = 1 by A2,FINSEQ_1:2,TARSKI:def 1; then
     (pre-Meas*FA).n = pre-Meas.(FA.1) by A4,FUNCT_1:12
        .= pre-Meas.A;
     hence thesis by A5;
    end;
    hence pre-Meas*<*A*> = <*pre-Meas.A*> by A3,FINSEQ_1:13;
end;
