theorem Th57: f|X +* g = f|(X\dom g) \/ g
proof
set f1=f|(X\dom g), a1=g;
dom f1 c= X\dom a1 & X\dom a1 misses dom a1 by
XBOOLE_1:79; then
A1: f1 tolerates a1 by PARTFUN1:56, XBOOLE_1:63;
f|X +* a1 = f|(X\dom a1\/(X/\dom a1)) +* a1 by Th48 .=
f1 +* f|(X/\dom a1) +* a1 by FUNCT_4:78 .=
f1 +* (f|(X/\dom a1) +*(a1 null {} null ({}\/dom a1)))
by FUNCT_4:14 .= f1 +* (f|X|dom a1 +* a1|(dom a1))
by RELAT_1:71 .= f1 +* (f|X +* a1)|(dom a1) by FUNCT_4:71
.= f1 +* a1 .= f1 \/ a1 by A1, FUNCT_4:30; hence thesis;
end;
