
theorem Th16:
for X be non empty set, f be PartFunc of X,REAL, E be set holds
 (R_EAL f)|E = R_EAL f|E
proof
    let X be non empty set, f be PartFunc of X,REAL, E be set;
    dom(R_EAL f|E) = dom(f|E) by MESFUNC5:def 7; then
A1: dom((R_EAL f)|E) = dom(R_EAL f|E) by MESFUNC5:def 7;
    now let x be Element of X;
     assume x in dom((R_EAL f)|E);
     (R_EAL f|E).x = (f|E).x by MESFUNC5:def 7;
     hence ((R_EAL f)|E).x = (R_EAL f|E).x by MESFUNC5:def 7;
    end;
    hence thesis by A1,PARTFUN1:5;
end;
