reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

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;
