let A, B be non empty set ; :: thesis: for C being set
for f, g being commuting Function st dom f c= Funcs A,(Funcs B,C) & rng f c= dom g holds
g * f = id (dom f)
let C be set ; :: thesis: for f, g being commuting Function st dom f c= Funcs A,(Funcs B,C) & rng f c= dom g holds
g * f = id (dom f)
let f, g be commuting Function; :: thesis: ( dom f c= Funcs A,(Funcs B,C) & rng f c= dom g implies g * f = id (dom f) )
assume A1: ( dom f c= Funcs A,(Funcs B,C) & rng f c= dom g ) ; :: thesis: g * f = id (dom f)
then A2: dom (g * f) = dom f by RELAT_1:46;

now

let x be set ; :: thesis: ( x in dom f implies (g * f) . x = x )
assume A3: x in dom f ; :: thesis: (g * f) . x = x
then reconsider X = x as Function by Def3;
A4: f . x in rng f by A3, FUNCT_1:def 5;
then reconsider Y = f . x as Function by A1, Def3;
thus (g * f) . x = g . (f . x) by A3, FUNCT_1:23
.= commute Y by A1, A4, Def3
.= commute (commute X) by A3, Def3
.= x by A1, A3, FUNCT_6:87 ; :: thesis: verum

end;

hence g * f = id (dom f) by A2, FUNCT_1:34; :: thesis: verum