reserve Y for RealNormSpace;
reserve X,Y for RealBanachSpace;
reserve Z for open Subset of REAL;
reserve a,b,c,d,e,r,x0 for Real;
reserve y0 for VECTOR of X;
reserve G for Function of X,X;

theorem Lm00:
for f be continuous PartFunc of REAL,the carrier of X
  st a in dom f holds integral(f,a,a) = 0.X
proof
   let f be continuous PartFunc of REAL,the carrier of X;
   assume A1: a in dom f;
   [.a,a.] = {a} by XXREAL_1:17; then
A2:[.a,a.] c= dom f by A1,ZFMISC_1:31;
A3:['a,a'] = [.a,a.] by INTEGRA5:def 3; then
A4:integral(f,['a,a']) = integral(f,a,a) by INTEGR18:16;
   vol ['a,a'] = a-a by INTEGRA6:5;
   hence thesis by A4,A2,A3,INTEGR18:17;
end;
