 reserve A for non empty Subset of REAL;
 reserve A for non empty closed_interval Subset of REAL;

theorem
  for a1,c,a2,d be Real, f be Function of REAL,REAL st
  c > 0 & d > 0 & a1 < a2 &
  f = ( d (#) TrapezoidalFS (a1-c,a1,a2,a2+c) ) | ['a1-c,a2+c']
    holds f is_integrable_on ['a1-c,a2+c']
proof
 let a1,c,a2,d be Real, f be Function of REAL,REAL;
 assume that
 A1: c > 0 & d > 0 & a1 < a2 and
 A2: f = ( d (#) TrapezoidalFS (a1-c,a1,a2,a2+c) ) | ['a1-c,a2+c'];
 a1-c < a1-0 & a2+0 < a2+c by A1,XREAL_1:15,XREAL_1:8; then
 (( d (#) TrapezoidalFS (a1-c,a1,a2,a2+c) ) | ['a1-c,a2+c'])
 | ['a1-c,a2+c'] is integrable by A1,Lm22d,INTEGRA5:def 1;
 hence thesis by INTEGRA5:def 1,A2;
end;
